maths basic

8/3/2019 Maths Basic

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THE REAL NUMBER SYSTEM

The real number system evolved over time by expanding the notion of what we mean bythe word number. At first, number meant something you could count, like how manysheep a farmer owns. These are called the natural numbers, or sometimes the countingnumbers.

NATURAL NUMBERS OR COUNTING NUMBERS1, 2, 3, 4, 5, . . .

The use of three dots at the end of the list is a common mathematical notation to indicatethat the list keeps going forever.

At some point, the idea of zero came to be considered as a number. If the farmer doesnot have any sheep, then the number of sheep that the farmer owns is zero. We call theset of natural numbers plus the number zero the whole numbers.

WHOLE NUMBERS

Natural Numbers together with zero0, 1, 2, 3, 4, 5, . . .

About the Number Zero

What is zero? Is it a number? How can the number of nothing be a number? Is zeronothing, or is it something?Well, before this starts to sound like a Zen koan, lets look at how we use thenumeral 0. Arab and Indian scholars were the first to use zero to develop theplace-value number system that we use today. When we write a number, we useonly the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These numerals can stand for

ones, tens, hundreds, or whatever depending on their position in the number. Inorder for this to work, we have to have a way to mark an empty place in a number,or the place values wont come out right. This is what the numeral 0 does. Thinkof it as an empty container, signifying that that place is empty. For example, thenumber 302 has 3 hundreds, no tens, and 2 ones.So is zero a number? Well, that is a matter of definition, but in mathematics wetend to call it a duck if it acts like a duck, or at least if its behavior is mostly duck-like. The number zero obeys most of the same rules of arithmetic that ordinarynumbers do, so we call it a number. It is a rather special number, though, because itdoesnt quite obey all the same laws as other numbersyou cant divide by zero,for example.

Note for math purists: In the strict axiomatic field development of the realnumbers, both 0 and 1 are singled out for special treatment. Zero is the additiveidentity, because adding zero to a number does not change the number. Similarly, 1is the multiplicative identity because multiplying a number by 1 does not change it.


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Even more abstract than zero is the idea of negative numbers. If, in addition to not havingany sheep, the farmer owes someone 3 sheep, you could say that the number of sheep thatthe farmer owns is negative 3. It took longer for the idea of negative numbers to beaccepted, but eventually they came to be seen as something we could call numbers. Theexpanded set of numbers that we get by including negative versions of the counting

numbers is called the integers.INTEGERS

Whole numbers plus negatives

. . . 4, 3, 2, 1, 0, 1, 2, 3, 4, . . .About Negative Numbers

How can you have less than zero? Well, do you have a checking account? Having lessthan zero means that you have to add some to it just to get it up to zero. And if you takemore out of it, it will be even further less than zero, meaning that you will have to add

even more just to get it up to zero.The strict mathematical definition goes something like this:For every real number n, there exists its opposite, denoted n, such that the sum of n and n is zero, or

n + ( n) = 0Note that the negative sign in front of a number is part of the symbol for that number:The symbol 3 is one object it stands for negative three, the name of the numberthat is three units less than zero.The number zero is its own opposite, and zero is considered to be neither negative norpositive.Read the discussion of subtraction for more about the meanings of the symbol .

The next generalization that we can make is to include the idea of fractions. While it isunlikely that a farmer owns a fractional number of sheep, many other things in real lifeare measured in fractions, like a half-cup of sugar. If we add fractions to the set ofintegers, we get the set of rational numbers.

RATIONAL NUMBERS

All numbers of the form , where a and b are integers (but b cannot be zero)

Rational numbers include what we usually call fractions

Notice that the word rational contains the word ratio, which should remind you offractions.

The bottom of the fraction is called the denominator. Think of it as the denominationittells you what size fraction we are talking about: fourths, fifths, etc.


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The top of the fraction is called the numerator. It tells you how many fourths, fifths, orwhatever.

RESTRICTION: The denominator cannot be zero! (But the numerator can)If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds

or zero fourths, than I dont have anything. However, it makes no sense at all to talkabout a fraction measured in zeroths.Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper fractions), or theycan be numbers bigger than 1 (called improper fractions), like two-and-a-half, which wecould also write as 5/2All integers can also be thought of as rational numbers, with a denominator of 1:

This means that all the previous sets of numbers (natural numbers, whole numbers, andintegers) are subsets of the rational numbers.Now it might seem as though the set of rational numbers would cover every possible

case, but that is not so. There are numbers that cannot be expressed as a fraction, andthese numbers are called irrational because they are not rational.Irrational NumbersCannot be expressed as a ratio of integers.As decimals they never repeat or terminate (rationals always do one or the other)

Examples:

Rational (terminates)

Rational (repeats)

Rational (repeats)

Rational (repeats)

Irrational (never repeats or terminates)

Irrational (never repeats or terminates)

More on Irrational Numbers

It might seem that the rational numbers would cover any possible number. Afterall, if I measure a length with a ruler, it is going to come out to some fraction


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maybe 2 and 3/4 inches. Suppose I then measure it with more precision. I will getsomething like 2 and 5/8 inches, or maybe 2 and 23/32 inches. It seems thathowever close I look it is going to be some fraction. However, this is not alwaysthe case.

Imagine a line segment exactly

one unit long:

Now draw another line one unitlong, perpendicular to the firstone, like this:

Now draw the diagonal connectingthe two ends:

Congratulations! You have just drawn a length that cannot be measured by anyrational number. According to the Pythagorean Theorem, the length of thisdiagonal is the square root of 2; that is, the number which when multiplied byitself gives 2.According to my calculator,

But my calculator only stops at eleven decimal places because it can hold nomore. This number actually goes on forever past the decimal point, without thepattern ever terminating or repeating.

This is because if the pattern ever stopped or repeated, you could write thenumber as a fractionand it can be proven that the square root of 2 can never bewritten as

for any choice of integers for a and b. The proof of this was considered quiteshocking when it was first demonstrated by the followers of Pythagoras 26


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centuries ago.THE REAL NUMBERS

Rationals + Irrationals

All points on the number lineOr all possible distances on the number line

When we put the irrational numbers together with the rational numbers, we finally havethe complete set of real numbers. Any number that represents an amount of something,such as a weight, a volume, or the distance between two points, will always be a realnumber. The following diagram illustrates the relationships of the sets that make up thereal numbers.

AN ORDERED SET

The real numbers have the property that they are ordered, which means that given anytwo different numbers we can always say that one is greater or less than the other. Amore formal way of saying this is:For any two real numbers a and b, one and only one of the following three statements istrue:1. a is less than b, (expressed as a < b)2. a is equal to b, (expressed as a = b)3. a is greater than b, (expressed as a > b)The Number LineThe ordered nature of the real numbers lets us arrange them along a line (imagine that theline is made up of an infinite number of points all packed so closely together that they

form a solid line). The points are ordered so that points to the right are greater than pointsto the left:

Every real number corresponds to a distance on the number line, starting at the center(zero). Negative numbers represent distances to the left of zero, and positive numbers aredistances to the right.


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The arrows on the end indicate that it keeps going forever in both directions.Absolute ValueWhen we want to talk about how large a number is without regard as to whether it ispositive or negative, we use the absolute value function. The absolute value of a numberis the distance from that number to the origin (zero) on the number line. That distance is

always given as a non-negative number.In short:

If a number is positive (or zero), the absolute value function does nothing to it:

If a number is negative, the absolute value function makes it positive:

WARNING: If there is arithmetic to do inside the absolute value sign, you must do it before taking the absolute valuethe absolute value function acts on the result ofwhatever is inside it. For example, a common error is

(WRONG)

The correct result is

ADDITION AND SUBTRACTION OF REAL NUMBERS

All the basic operations of arithmetic can be defined in terms of addition, so we will takeit as understood that you have a concept of what addition means, at least when we aretalking about positive numbers.Addition on the Number LineA positive number represents a distance to the right on the number line, starting fromzero (zero is also called the origin since it is the starting point). When we add another

positive number, we visualize it as taking another step to the right by that amount. Forexample, we all know that 2 + 3 = 5. On the number line we would imagine that we startat zero, take two steps to the right, and then take three more steps to the right, whichcauses us to land on positive 5.

ADDITION OF NEGATIVE NUMBERS

What does it mean to add negative numbers? We view a negative number as a

displacement to the left on the number line, so we follow the same procedure as beforebut when we add a negative number we take that many steps to the left instead of theright.

Examples:2 + (3) = 1

First we move two steps to the right, and then three steps to the left:


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(2) + 3 = 1

We move two steps to the left, and then three steps to the right:

(2) + (3) = 5

Two steps to the left, and then three more steps to the left:

From these examples, we can make the following observations:1. If we add two positive numbers together, the result will be positive2. If we add two negative numbers together, the result will be negative3. If we add a positive and a negative number together, the result could be positive ornegative, depending on which number represents the biggest step.Subtraction

There are two ways to define subtraction: by a related addition statement, or as adding theopposite.Subtraction as Related Additiona b = c if and only if a = b + cSubtraction as Adding the OppositeFor every real number b there exists its opposite b, and we can define subtraction asadding the opposite:a b = a + (b) In algebra it usually best to always think of subtraction as adding the oppositeDistinction Between Subtraction and NegationThe symbol means two different things in math. If it is between two numbers itmeans subtraction, but if it is in front of one number it means the opposite (or negative)of that number.Subtraction is binary (acts on two numbers), but negation is unary (acts on only onenumber).

Calculators have two different keys to perform these functions. The key with aplain minus sign is only for subtraction:

Negation is performed by a key that looks like one of these:


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Remember that subtraction can always be thought of as adding the opposite. In fact, wecould get along just fine without ever using subtraction.Subtraction on the Number Line

Addition of a positive number moves to the right, and adding a negative moves to the left.Subtraction is just the opposite: Subtraction of a positive number moves to the left, and

subtracting a negative moves to the right. Notice that subtracting a negative is the same thing as adding a positive.

Fractions

Fractions, also called rational numbers, are numbers of the form , where a and b areintegers (but b cannot be zero).The bottom number is called the denominator. Think of it as the denomination: it tells

you what size units you are talking aboutfourths, fifths, or whatever.The top number is the numerator. It tells you how many of those units you have. Forexample, if I have 3 quarters in my pocket, then I have three-fourths of a dollar. Thedenomination is quarters (fourths), and I have three of them: 3/4.Improper FractionsOrdinarily we think of fractions as being between zero and one, like 3/4 or 2/3. These arecalled proper fractions. In these fractions, the numerator is smaller than the denominatorbut there is no reason why we can not have a numerator bigger than the denominator.Such fractions are called improper.What does an improper fraction like 5/4 mean? Well, if we have 5 quarters of somethingthen we have more than one whole of that something. In fact, we have one whole plusone more quarter (if you have 5 quarters in change, you have a dollar and a quarter).Mixed Number Notation

One way of expressing the improper fraction 5/4 is as the mixed number , which isread as one and one-fourth. This notation is potentially confusing and is not advised inalgebra.One cause of confusion is that in algebra we use the convention that multiplication isimplied when two quantities are written next to each other with no symbols in between.However, the mixed number notation implies addition, not multiplication. For example,

means 1 plus one-quarter, and 3 1/2 = 3 + 1/2.It is possible to do arithmetic with mixed numbers by treating the whole number partsand the fractional parts separately, but it is generally more convenient in algebra toalways write improper fractions. When you encounter a problem with mixed numbers,the first thing you should do is convert them to improper fractions.

Conversion

Mixed Number to Improper Fraction


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A. Multiply the integer part with the bottom of the fraction part.B. Add the result to the top of the fraction.The general formula is

Improper Fraction to Mixed Number

Do the division to get the integer partPut the remainder over the old denominator to get the fractional part.Multiplying, Reducing, and Dividing FractionsEquivalent FractionsEquivalent fractions are fractions that have the same value, for example

etc.Although all these fractions are written differently, they all represent the same quantity.You can measure a half-cup of sugar or two quarter-cups of sugar, or even four eighth-cups of sugar, and you will still have the same amount of sugar.

Multiply by a form of One

A fraction can be converted into an equivalent fraction by multiplying it by a form of 1.

The number 1 can be represented as a fraction because any number divided by itself isequal to 1 (remember that the fraction notation means the same thing as division). Inother words,

etc.Now if you multiply a number by 1 it does not change its value, so if we multiply afraction by another fraction that is equal to 1, we will not be changing the value of theoriginal fraction. For example,

In this case, 2/3 represents exactly the same quantity as 4/6, because all we did was tomultiply 2/3 by the number 1, represented as the fraction 2/2.Multiplying the numerator and denominator by the same number to produce anequivalent fraction is called building up the fraction.


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Reduced Form

Numerator and Denominator Have No Common Factors

Procedure:1. Write out prime factorization of Numerator and Denominator2. Cancel all common factors

This procedure is just the opposite of building up a fraction by multiplying it by a fractionequivalent to 1.

Prime FactorsA number is prime if it has no whole number factors other than 1 times itself, thatis, the number cannot be written as a product of two whole numbers (except 1times itself).Example: 6 is not prime because it can be written as 2 3Example: 7 is prime because the only way to write it as a product of wholenumbers is 1 7The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, . . .

There are an infinite number of prime numbers (the list goes on forever).Any non-prime number can be decomposed into a product of prime numbersExample: 4 = 2 2Example: 12 = 2 2 3The Branching MethodThis method works well for larger numbers that might have many factors. All youneed to do is think of any two numbers that multiply to give your original number,and write them below it. Continue this process for each number until each branchends in a prime number. The factors of the original number are the prime numberson the ends of all the branches.

Example: Factor the number

60

60 = 2 2 3 5

Notes

Notice how I started with the smallest numbers: first 2s, then 3s, and so on. Thisis not required but it keeps the result nicely in order.If a number is even, then it is divisible by 2.If a number ends in 0 or 5, then it is divisible by 5.If the digits of a number add up to a number divisible by 3, then the number isdivisible by 3. In this example 15 gives 1 + 5 = 6, which is divisible by 3, andtherefore 15 is divisible by 3.Large numbers with large prime factors are notoriously hard to factorit is


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mainly just a matter of trial and error. The public-key encryption system forsending secure computer data uses very large numbers that need to be factored inorder to break the code. The code is essentially unbreakable because it would takean enormous amount of computer time to try every possible prime factor.

Multiplying Fractions

Multiply Numerators and DenominatorsExample:

And reduce result if needed

Canceling common factors first makes multiplication easierIf you dont reduce the factors before multiplying, the answer will have to be reduced.Example:

Remember that canceling always leaves a 1 behind, because you are really dividing thenumerator and the denominator by the same number.

Adding and Subtracting Fractions Add Numerators when Denominators Are the Same

If the denominators are not the same, make them the same by building up thefractions so that they both have a common denominator. Any common denominator will work, but the answer will have to be reduced if it isnot the Least Common Denominator. The product of all the denominators is always a common denominator (but notnecessarily the Least Common Denominator).Least Common Denominator (LCD)By InspectionThe smallest number that is evenly divisible by all the denominatorsIn GeneralThe LCD is the product of all the prime factors of all the denominators, each factor takenthe greatest number of times that it appears in any single denominator.Example:


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Factor the denominators:

Assemble LCD:

Note that the three only appears once, because it is only needed once to make either the12 or the 15:

Now that you have found the LCD, multiply each fraction (top and bottom) by whateveris needed to build up the denominator to the LCD:

Then add the numerators and reduce if needed (using the LCD does not guarantee thatyou wont have to reduce):

Properties of Real NumbersThe following table lists the defining properties of the real numbers (technically calledthe field axioms). These laws define how the things we call numbers should behave.

Addition MultiplicationCommutative

For all real a, ba + b = b + a

Commutative

For all real a, bab = ba

AssociativeFor all real a, b, ca + (b + c) = (a + b) + c

AssociativeFor all real a, b, c(ab)c = a(bc)

IdentityThere exists a real number 0 suchthat for every real a

IdentityThere exists a real number 1 suchthat for every real a


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a + 0 = a a 1 = aAdditive Inverse(Opposite)For every real number a thereexist a real number, denoted (a),

such thata + (a) = 0

Multiplicative Inverse (Reciprocal)For every real number a except 0there exist a real number, denoted

, such that

a = 1Distributive LawFor all real a, b, ca(b + c) = ab + ac, and (a + b)c = ac + bc

The commutative and associative laws do not hold for subtraction or division:a b is not equal to b aa b is not equal to b aa (b c) is not equal to (a b) c

a (b c) is not equal to (a b) cTry some examples with numbers and you will see that they do not work.What these laws mean is that order and grouping don't matter for addition andmultiplication, but they certainly do matter for subtraction and division. In this way,addition and multiplication are cleaner than subtraction and division. This will becomeimportant when we start talking about algebraic expressions. Often what we will want todo with an algebraic expression will involve rearranging it somehow. If the operations areall addition and multiplication, we don't have to worry so much that we might bechanging the value of an expression by rearranging its terms or factors. Fortunately, wecan always think of subtraction as an addition problem (adding the opposite), and we canalways think of division as a multiplication (multiplying by the reciprocal).

You may have noticed that the commutative and associative laws read exactly the sameway for addition and multiplication, as if there was no difference between them otherthan notation. The law that makes them behave differently is the distributive law, becausemultiplication distributes over addition, not vice-versa.. The distributive law is extremelyimportant, and it is impossible to understand algebra without being thoroughly familiarwith this law.Example: 2(3 + 4)According to the order of operations rules, we should evaluate this expression by firstdoing the addition inside the parentheses, giving us2(3 + 4) = 2(7) = 14But we can also look at this problem with the distributive law, and of course still get the

same answer. The distributive law says that

SURDSSurds are numbers left in 'square root form' (or 'cube root form' etc). They are thereforeirrational numbers. The reason we leave them as surds is because in decimal form theywould go on forever and so this is a very clumsy way of writing them.Addition and Subtraction of Surds
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Adding and subtracting surds are simple- however we need the numbers being squarerooted (or cube rooted etc) to be the same.47 - 27 = 27.52 + 82 = 132

Note: 52 + 33 cannot be manipulated because the surds are different (one is 2 and oneis 3).Multiplication5 15 = 75 (= 15 5)= 25 3= 53 .

(1 + 3) (2 - 8) [Thebrackets are expanded as usual]= 2 - 8 + 23 - 24= 2 - 22 + 23 - 26Rationalising the Denominator

It is untidy to have a fraction which has a surd denominator. This can be 'tidied up' bymultiplying the top and bottom of the fraction by a particular expression. This is knownas rationalising the denominator, since surds are irrational numbers and so you arechanging the denominator from an irrational to a rational number.ExampleRationalise the denominator of:a) 1

2 .

b) 1 + 21 - 2

a) Multiply the top and bottom of the fraction by 2. The top will become 2 and thebottom will become 2 (2 times 2 = 2).

b) In situations like this, look at the bottom of the fraction (the denominator) and changethe sign (in this case change the plus into minus). Now multiply the top and bottom of thefraction by this.

Therefore:1 + 2 = (1 + 2)(1 + 2) = 1 + 2 + 2 + 22 = 3 + 32

1 - 2 (1 - 2)(1 + 2) 1 + 2 - 2 - 2 - 1

= -3(1 + 2)FactorisingExpanding BracketsBrackets should be expanded in the following ways:For an expression of the form a(b + c), the expanded version is ab + ac, i.e., multiply theterm outside the bracket by everything inside the bracket (e.g. 2x(x + 3) = 2x + 6x
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[remember x x is x]).For an expression of the form (a + b)(c + d), the expanded version is ac + ad + bc + bd, inother words everything in the first bracket should be multiplied by everything in thesecond.Example

Expand (2x + 3)(x - 1):(2x + 3)(x - 1)= 2x - 2x + 3x - 3= 2x + x - 3FactorisingFactorising is the reverse of expanding brackets, so it is, for example, putting 2x + x - 3into the form (2x + 3)(x - 1). This is an important way of solving quadratic equations.The first step of factorising an expression is to 'take out' any common factors which theterms have. So if you were asked to factorise x + x, since x goes into both terms, youwould write x(x + 1) .Factorising Quadratics

There is no simple method of factorising a quadratic expression, but with a little practiseit becomes easier. One systematic method, however, is as follows:ExampleFactorise 12y - 20y + 3= 12y - 18y - 2y + 3 [here the 20y has been split up into two numbers whose multipleis 36. 36 was chosen because this is the product of 12 and 3, the other two numbers].The first two terms, 12y and -18y both divide by 6y, so 'take out' this factor of 6y.6y(2y - 3) - 2y + 3 [we can do this because 6y(2y - 3) is the same as 12y - 18y]Now, make the last two expressions look like the expression in the bracket:6y(2y - 3) -1(2y - 3)The answer is (2y - 3)(6y - 1)ExampleFactorise x + 2x - 8We need to split the 2x into two numbers which multiply to give -8. This has to be 4 and-2.x + 4x - 2x - 8x(x + 4) - 2x - 8x(x + 4)- 2(x + 4)(x + 4)(x - 2)

Once you work out what is going on, this method makes factorising any expression easy.It is worth studying these examples further if you do not understand what is happening.Unfortunately, the only other method of factorising is by trial and error.The Difference of Two SquaresIf you are asked to factorise an expression which is one square number minus another,you can factorise it immediately. This is because a - b = (a + b)(a - b) .ExampleFactorise 25 - x= (5 + x)(5 - x) [imagine that a = 5 and b = xAlgebraic Fractions


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Algebraic FractionsAlgebraic fractions are simplyfractions with algebraic expressions on the top and/orbottom.When adding or subtracting algebraic fractions, the first thing to do is to put them onto acommon denominator (by cross multiplying).

e.g. 1 + 4(x + 1) (x + 6)

= 1(x + 6) + 4(x + 1)(x + 1)(x + 6)

= x + 6 + 4x + 4(x + 1)(x + 6)

= 5x + 10

(x + 1)(x + 6)Solving equationsWhen solving equations containing algebraic fractions, first multiply both sides by anumber/expression which removes the fractions.ExampleSolve 10 -2 = 1

(x + 3) x

multiply both sides by x(x + 3): 10x(x + 3) - 2x(x + 3) = x(x + 3)

(x + 3) x

10x - 2(x + 3) = x2 + 3x [aftercancelling] 10x - 2x - 6 = x2 + 3x x2 - 5x + 6 = 0 (x - 3)(x - 2) = 0 either x = 3 or x = 2Solving EquationsSee Also: Quadratic Equations and Simultaneous EquationsTrial and ImprovementAny equation can be solved by trial and improvement (/error). However, this is a tediousprocedure. Start by estimating the solution (you may be given this estimate). Then

substitute this into the equation to determine whether your estimate is too high or too low.Refine your estimate and repeat the process.ExampleSolve t + t = 17 by trial and improvement.

Firstly, select a value of t to try in the equation. I have selected t = 2. Put this value intothe equation. We are trying to get the answer of 17.If t = 2, then t + t = 2 + 2 = 10 . This is lower than 17, so we try a higher value for t.
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If t = 2.5, t + t = 18.125 (too high)If t = 2.4, t + t = 16.224 (too low)If t = 2.45, t + t = 17.156 (too high)If t = 2.44, t + t = 16.966 (too low)If t = 2.445, t + t = 17.061 (too high)

So we know that t is between 2.44 and 2.445. So to 2 decimal places, t = 2.44.IterationThis is a way of solving equations. It involves rearranging the equation you are trying tosolve to give an iteration formula. This is then used repeatedly (using an estimate to startwith) to get closer and closer to the answer.An iteration formula might look like the following (this is for the equation x2 = 2x + 1):xn+1 = 2 + 1

xn

You are usually given a starting value, which is called 0. If x0 = 3, substitute 3 into the

original equation where it says xn. This will give you x1. (This is because if n = 0, x1 = 2 +1/x0 and x0 = 3).x1 = 2 + 1/3 = 2.333 333 (by substituting in 3).To find x2, substitute the value you found for x1.x2 = 2 + 1/(2.333 333) = 2.428 571

Repeat this until you get an answer to a suitable degree of accuracy. This may be aboutthe 5th value for an answer correct to 3s.f. In this example, x5 = 2.414...Examplea) Show that x = 1 + 11

x - 3

is a rearrangement of the equation x - 4x - 8 = 0.

b) Use the iterative formula:xn+1 = 1 + 11

xn - 3together with a starting value of x1 = -2 to obtain a root of the equation x - 4x - 8 = 0accurate to one decimal place.

a) multiply everything by (x - 3):x(x - 3) = 1(x - 3) + 11so x - 3x = x + 8

so x - 4x - 8 = 0

b) x1 = -2x2 = 1 + 11 (substitute -2 into the iteration formula)

-2 - 3= -1.2


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x3 = 1 + 11 (substitute -1.2 into the above formula)-1.2 - 3

= -1.619x4 = -1.381x5 = -1.511

x6 = -1.439x7 = -1.478therefore, to one decimal place, x = 1.5 .Quadratic EquationsA quadratic equation is an equation where the highest power of x is x2. There are variousmethods of solving quadratic equations, as shown below.

Important point about square roots: 62 = 36. But also (-6)2 = 36 because -6 -6 = + 36 (aminus a minus = a plus). Therefore there are two square roots of 36: +6 and -6. We call6 the positive square root of 36 and -6 is called the negative square root of 36.So if x2 = 36, then x = +6 or -6 (since squaring either of these numbers will give 36).

However, the square root sign means "positive square root". So 36 = + 6 (only).Completing the Square9 and 25 can be written as 32 and 52 whereas 7 and 11 cannot be written as the square ofanother exact number. 9 and 25 are called perfect squares. Another example is (9/4) =(3/2)2. In a similar way, x2 + 2x + 1 = (x + 1)2.To make x2 + 6x into a perfect square, we add (62/4) = 9. The resulting expression, x2 +6x + 9 = (x + 3)2 and so is a perfect square. This is known as completing the square. Tocomplete the square in this way, we take the number before the x, square it, and divide itby 4. This technique can be used to solve quadratic equations, as demonstrated in thefollowing example.Example

Solve x2

- 6x + 2 = 0 by completing the squarex2 - 6x = -2[To complete the square on the LHS (left hand side), we must add 62/4 = 9. We must, ofcourse, do this to the RHS also]. x2 - 6x + 9 = 7 (x - 3)2 = 7[Now take the square root of each side] x - 3 = 2.646 (the square root of 7 is +2.646 or -2.646) x = 5.646 or 0.354

Completing the square can also be used to find the maximum or minimum point on a

graph.ExampleFind the minimum of the graph y = 3x2 - 6x - 3 .

In this case, the x2 has a '3' in front of it, so we start by taking the three out: y = 3(x2 - 2x-1) . [This is the same since multiplying it out gives 3x2 - 6x - 3]Now complete the square for the bit in the bracket: y = 3[(x - 1)2 - 2]


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Multiply out the big bracket: y = 3(x - 1)2 - 6

We are trying to find the minimum value that this graph can be. (x - 1)2 must be zero orpositive, since squaring a number always gives a positive answer. So the minimum value

will occur when (x - 1)2

= 0, which is when x = 1. When x = 1, y = -6 . So the minimumpoint is at (1, -6).

Some people don't like the method of completing the square to solve equations and analternative is to use the quadratic formula. This is actually derived by completing thesquare.

The Quadratic Formula

Where the equation is ax2 + bx + c = 0ExampleSolve 3x2 + 5x - 8 = 0

x = -5 ( 52 - 43(-8))6

= -5 (25 + 96)6

= -5 (121)6

= -5 + 11 or -5 - 116 6

x = 1 or -2.67FactorisingSometimes, quadratic equations can be solved by factorising. In this case, factorising isprobably the easiest way to solve the equation.ExampleSolve x2 + 2x - 8 = 0 (x - 2)(x + 4) = 0 either x - 2 = 0 or x + 4 = 0 x = 2 or x = - 4

If you do not understand the third line, remember that for (x - 2)(x + 4) to equal zero,then one of the two brackets must be zero.Sin, Cos and Tan


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A right-angled triangle is a triangle in which one of the angles is a right-angle. Thehypotenuse of a right angled triangle is the longest side, which is the one opposite theright angle. The adjacent side is the side which is between the angle in question and theright angle. The opposite side is opposite the angle in question.In any right angled triangle, for any angle:

The sine of the angle = the length of the opposite sidethe length of the hypotenuse

The cosine of the angle = the length of the adjacent side

the length of the hypotenuse

The tangent of the angle = the length of the opposite sidethe length of the adjacent side

So in shorthand notation:sin = o/h cos = a/h tan = o/aOften remembered by: soh cah toaExampleFind the length of side x in the diagram below:


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The angle is 60 degrees. We are given the hypotenuse and need to find the adjacent side.This formula which connects these three is:cos(angle) = adjacent / hypotenusetherefore, cos60 = x / 13therefore, x = 13 cos60 = 6.5therefore the length of side x is 6.5cm.

The Graphs of Sin, Cos and TanThe following graphs show the value of sin, cos and tan against ( represents anangle). From the sin graph we can see that sin = 0 when = 0 degrees, 180 degrees and360 degrees.


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Note that the graph of tan has asymptotes (lines which the graph gets close to, but nevercrosses). These are the red lines (they aren't actually part of the graph).Also notice that the graphs of sin, cos and tan are periodic. This means that they repeatthemselves. Therefore sin() = sin(360 + ), for example.


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Notice also the symmetry of the graphs. For example, cos is symmetrical in the y-axis,which means that cos = cos(-). So, for example, cos(30) = cos(-30).Also, sin x = sin (180 - x) because of the symmetry of sin in the line = 90.

Summary of trigonometric identitiesYou have seen quite a few trigonometric identities in the past few pages. It is convenientto have a summary of them for reference. These identities mostly refer to one angledenoted t, but there are a few of them involving two angles, and for those, the other angleis denoted s..More important identitiesYou don't have to know all the identities off the top of your head. But these you should.Defining relations for tangent, cotangent, secant, and cosecant in terms of sine andcosine.

tan t =sin t

cos t

cot t =1

tan t

=cos t

sin tsec t =

1cos t

csc t =1sin t

The Pythagorean formula for sines and cosines.sin2 t + cos2 t = 1Identities expressing trig functions in terms of their complementscos t = sin( /2 t) sin t = cos( /2 t)cot t = tan( /2 t) tan t = cot( /2 t)csc t = sec( /2 t) sec t = csc( /2 t)Periodicity of trig functions. Sine, cosine, secant, and cosecant have period 2 whiletangent and cotangent have period .

sin (t + 2 ) = sin tcos (t + 2 ) = cos ttan (t + ) = tan tIdentities for negative angles. Sine, tangent, cotangent, and cosecant are odd functionswhile cosine and secant are even functions.sin t = sin tcos t = cos ttan t = tan tSum formulas for sine and cosinesin (s + t) = sin s cos t + cos s sin tcos (s + t) = cos s cos t sin s sin t

Double angle formulas for sine and cosinesin 2t = 2 sin t cos tcos 2t = cos2 t sin2 t = 2 cos2 t 1 = 1 2 sin2 tLess important identitiesYou should know that there are these identities, but they are not as important as thosementioned above. They can all be derived from those above, but sometimes it takes a bitof work to do so.The Pythagorean formula for tangents and secants.


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sec2 t = 1 + tan2 tIdentities expressing trig functions in terms of their supplementssin( t) = sin tcos( t) = cos ttan( t) = tan t

Difference formulas for sine and cosinesin (s t) = sin s cos t cos s sin tcos (s t) = cos s cos t + sin s sin tSum, difference, and double angle formulas for tangent

tan (s + t) =tan s + tan t1 tan s tan t

tan (s t) =tan s tan t1 + tan s tan t

tan 2t =2 tan t1 tan2 t

Half-angle formulas

sin t/2 = ((1 cos t) / 2)cos t/2 = ((1 + cos t) / 2)

tan t/2 =sin t1 + cos t

=1 cos tsin t

Truly obscure identitiesThese are just here for perversity. Yes, of course, they have some applications, but they'reusually narrow applications, and they could just as well be forgotten until, if ever,needed.Product-sum identities

sin s + sin t = 2 sins + t2

coss t2

sin s sin t = 2 cos s + t2

sin s t2

cos s + cos t = 2 coss + t2

coss t2

cos s cos t = 2 sins + t2

sins t2

Product identities

sin s cos t =sin (s + t) + sin (s t)2

cos s cos t =cos (s + t) + cos (s t)2

sin s sin t =cos (s t) cos (s +t)2

Aside: weirdly enough, these productidentities were used before logarithms toperform multiplication. Here's how you

could use the second one. If you want tomultiply x times y, use a table to look up theangle s whose cosine is x and the angle twhose cosine is y. Look up the cosines of thesum s + t, and the difference s t. Averagethose two cosines. You get the product xy!Three table look-ups, and computing a sum,a difference, and an average rather than one


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multiplication. Tycho Brahe (1546-1601),among others, used this algorithm known asprosthaphaeresis.

Triple angle formulas. You can easily reconstruct these from the addition and doubleangle formulas.

sin 3t = 3 sin t 4 sin3

tcos 3t = 4 cos3 t 3 cos t

tan 3t =3 tan t tan3t1 3 tan2t

More half-angle formulas. (These are used in calculus for a particular kind of substitutionin integrals sometimes called the Weierstrass t-substitution.)

sin t =2 tan t/21 + tan2 t/2

cos t =1 tan2 t/21 + tan2 t/2

tan t =2 tan t/21 tan2 t/2

Top of Form

Bottom of Form

1996, 1997.David E. JoyceDepartment of Mathematics and Computer ScienceClark UniversityWorcester, MA 01610

Dave's Short Trig Course is located at http://www.clarku.edu/~djoyce/java/trig

VariablesWhat is a variable? It is a box, and it exists to contain a value. Sometimes the value is alreadyinside the box, and you have to figure out what that value is. Other times, the box is empty, andyou get to pick the value to put inside. More about that later. First...Remember when you were in elementary school, and you were learning your addition? Theteacher would hand you worksheets that said things like:

; fill in the box.Variables are the same thing. Now we say:

x + 2 = 5 ; solve forx.Why did we switch from boxes to letters? Because letters are better. Boxes come in only a fewshapes, but letters come in many varieties, and letters can stand for something. For instance, theformula from geometry for finding a circle's circumference is:This formula makes more sense than, say:

The two formulae say exactly the same thing, but using "C" for "circumference" and "r" for"radius" is more useful than using "square" and "triangle, respectively. Boxes are fine, but lettersare better.

In the above discussion, I illustrated both uses of variables. In the equation "x + 2 = 5", x canonly have a value of3. The statement (the equation) is not true for any other value. That is to say,the value ofx is "fixed"; we just have to figure out what it is.
http://www.clarku.edu/~djoyce/trig/copyright.htmlhttp://www.clarku.edu/~djoyce/trig/copyright.htmlhttp://www.clarku.edu/~djoyce/http://www.clarku.edu/departments/mathcs/http://www.clarku.edu/departments/mathcs/http://www.clarku.edu/http://www.clarku.edu/~djoyce/trig/copyright.htmlhttp://www.clarku.edu/~djoyce/http://www.clarku.edu/departments/mathcs/http://www.clarku.edu/


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On the other hand, in the equation " ", the radius rcan be any non-negative number wechoose we get to pick! and then we get to figure out what the circumference C is. In the firstcase, we had to open the box to see what was already inside; in the second, we got to put thevalue in ourselves.

Now that we have variables, what do we do with them? Go back in your mind again to elementary

school: Your teacher would have you add "2 apples plus 6 apples is 8 apples". The same rulesapply to variables: "2 boxes plus 6 boxes is 8 boxes", or, using variables, "2x + 6x = 8x". "A boxand another box is two boxes", or "x + x = 2x". "Two dollars, less the ten that you owe to yourfriend, means that you're eight dollars in the red", or "2x 10x = 8x".But note: "2 apples plus 6 oranges" is just 2 apples and 6 oranges; they might make a nice fruitsalad, but they're not 8 of anything. In the same way, "2x + 6y" is just 2x + 6y; you can'tcombine the two variables into one. Copyright Elizabeth Stapel 1999-2009 All Rights Reserved

When multiplying, we use exponents. For instance, (5)(5) = 52. Of course, we can simplify thisas 52 = 25. Similarly, (x)(x) = x2. But, until we know what value to put in forx, we cannotsimplify this.

Don't confuse multiplication and addition: (x)(x) does not equal 2x, just as (5)(5) does not equal(2)(5); instead, (x)(x) equals x2. (Note the technique I just used: If you're not sure what to do

with the variables, put in numbers, where you know what to do. Then, whatever you did with thenumbers, do that with the variables.)

Exponents: Basic Rules (page 1 of 5)Sections: Basics, Negative exponents,Scientific notation, Engineering notation,Fractionalexponents

Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the

shorthand for multiplying three copies of the number5 three is shown on the right-hand side of ofthe "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for howevermany times the value is being multiplied. The thing that's being multiplied, being 5 in thisexample, is called the "base".This process of using exponents is called "raising to a power", where the exponent is the "power".

The expression "53" is pronounced as "five, raised to the third power" or "five to the third". Thereare two specially-named powers: "to the second power" is generally pronounced as "squared",and "to the third power" is generally pronounced as "cubed". So "53" is commonly pronounced as"five cubed".

When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "33".But with variables, we need the exponents, because we'd rather deal with "x6" than with"xxxxxx".Exponents have a few rules that we can use for simplifying expressions.

Simplify (x3)(x4) Copyright Elizabeth Stapel 1999-2009 All Rights ReservedTo simplify this, I can think in terms of what those exponents mean. "To the third" means"multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can"expand" the two factors, and then work backwards to the simplified form:

(x3)(x4) = (xxx)(xxxx)= xxxxxxx= x7

Note that x7 also equals x(3+4). This demonstrates the first basic exponent rule: Whenever youmultiply two terms with the same base, you can add the exponents:

( x m ) ( x n ) = x( m + n )

However, we can NOT simplify (x4)(y3), because the bases are different: (x4)(y3) = xxxxyyy =(x4)(y3). Nothing combines.Simplify (x2)4
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Just as with the previous exercise, I can think in terms of what the exponents mean. The "to the

fourth" means that I'm multiplying four copies ofx2:(x2)4 = (x2)(x2)(x2)(x2)

= (xx)(xx)(xx)(xx)= xxxxxxxx= x

8

Note that x(8 also equals x( 24 ). This demonstrates the second exponent rule: Whenever youhave an exponent expression that is raised to a power, you can multiply the exponent and power:

( xm ) n = x m n

If you have a product inside parentheses, and a power on the parentheses, then the power goes

on each element inside. For instance, (xy2)3 = (xy2)(xy2)(xy2) = (xxx)(y2y2y2) = (xxx)(yyyyyy) = x3y6 = (x)3(y2)3. Another example would be:

Warning: This rule does NOT work if you have a sum or difference within the parentheses.Exponents, unlike mulitiplication, do NOT "distribute" over addition.

For instance, given (3 + 4)2

, do NOT succumb to the temptation to say "This equals 32

+ 42

= 9+ 16 = 25", because this is wrong. Actually, (3 + 4)2 = (7)2 = 49, not 25. When in doubt, writeout the expression according to the definition of the power. Given (x 2)2, don't try to do this inyour head. Instead, write it out: "squared" means "times itself", so (x 2)2 = (x 2)(x 2) = xx 2x 2x + 4 = x2 4x + 4.The mistake of erroneously trying to "distribute" the exponent is most often made when thestudent is trying to do everything in his head, instead of showing his work. Do things neatly, andyou won't be as likely to make this mistake.There is one other rule that may or may not be covered at this stage:

Anything to the power zero is just "1".This rule is explained on the next page. In practice, though, this rule means that some exercisesmay be a lot easier than they may at first appear:

Simplify [(3x

4

y

7

z

12

)

5

(5x

9

y

3

z

4

)

2

]

0

Who cares about that stuff inside the square brackets? I don't, because the zero power on theoutside means that the value of the entire thing is just 1.Sections: Polynomial basics,Combining "like terms"

By now, you should be familiar with variables and exponents, and you may have dealt with

expressions like 3x4 or6x. Polynomials are sums of these "variables and exponents"expressions. Each piece of the polynomial, each part that is being added, is called a "term".Polynomial terms have variables which are raised to whole-number exponents (or else the termsare just plain numbers); there are no square roots of variables, no fractional powers, and novariables in the denominator of any fractions. Here are some examples:

6x 2 This is NOTa polynomial term...

...because the variable hasa negative exponent.

1/x2This is NOTa polynomial term...

...because the variable is inthe denominator.

sqrt(x) This is NOTa polynomial term...

...because the variable isinside a radical.

4x2 This IS a polynomial term... ...because it obeys all therules.
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Here is a typical polynomial:

Notice the exponents on the terms. The first term has an exponent of2; the second term has an"understood" exponent of1; and the last term doesn't have any variable at all. Polynomials areusually written this way, with the terms written in "decreasing" order; that is, with the largestexponent first, the next highest next, and so forth, until you get down to the plain old number.Any term that doesn't have a variable in it is called a "constant" term because, no matter what

value you may put in for the variable x, that constant term will never change. In the picture above,no matter what x might be, 7 will always be just 7.The first term in the polynomial, when it is written in decreasing order, is also the term with thebiggest exponent, and is called the "leading term".The exponent on a term tells you the "degree" of the term. For instance, the leading term in theabove polynomial is a "second-degree term" or "a term of degree two". The second term is a "firstdegree" term. The degree of the leading term tells you the degree of the whole polynomial; thepolynomial above is a "second-degree polynomial". Here are a couple more examples:Give the degree of the following polynomial: 2x5 5x3 10x + 9This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degreeterm, and a constant term.This is a fifth-degree polynomial.

Give the degree of the following polynomial: 7x4 + 6x2 + xThis polynomial has three terms, including a fourth-degree term, a second-degree term, and afirst-degree term. There is no constant term.

This is a fourth-degree polynomial.When a term contains both a number and a variable part, the number part is called the"coefficient". The coefficient on the leading term is called the "leading" coefficient.

In the above example, the coefficient of the leading term is 4; the coefficient of the second term is

3; the constant term doesn't have a coefficient. Copyright Elizabeth Stapel 2006-2008 All Rights ReservedThe "poly" in "polynomial" means "many". I suppose, technically, the term "polynomial" shouldonly refer to sums of many terms, but the term is used to refer to anything from one term to thesum of a zillion terms. However, the shorter polynomials do have their own names:a one-term polynomial, such as 2x or4x2, may also be called a "monomial" ("mono" meaning"one")a two-term polynomial, such as 2x + y orx2 4, may also be called a "binomial" ("bi" meaning"two")


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a three-term polynomial, such as 2x + y + z orx4 + 4x2 4, may also be called a "trinomial"("tri" meaning "three")I don't know if there are names for polynomials with a greater numbers of terms; I've never heardof any names other than what I've listed.

Polynomials are also sometimes named for their degree:a second-degree polynomial, such as 4x2, x2 9, orax2 + bx + c, is also called a "quadratic"a third-degree polynomial, such as 6x3 orx3 27, is also called a "cubic"a fourth-degree polynomial, such as x4 or2x4 3x2 + 9, is sometimes called a "quartic"a fifth-degree polynomial, such as 2x5 orx5 4x3 x + 7, is sometimes called a "quintic"There are names for some of the higher degrees, but I've never heard of any names being usedother than the ones I've listed.By the way, yes, "quad" generally refers to "four", as when an ATV is referred to as a "quad bike".For polynomials, however, the "quad" from "quadratic" is derived from the Latin for "makingsquare". As in, if you multiply length by width (of, say, a room) to find the area in "square" units,the units will be raised to the second power. The area of a room that is 6 meters by 8 meters is 48m2. So the "quad" refers to the four corners of a square, from the geometrical origins of parabolasand early polynomials.

Evaluation"Evaluating" a polynomial is the same as evaluatinganything else: you plug in the given value of

x, and figure out what y is supposed to be. For instance:Evaluate 2x3 x2 4x + 2 at x = 3I need to plug in "3" for the "x", remembering to be careful with my parentheses and thenegatives:

2(3)3 (3)2 4(3) + 2= 2(27) (9) + 12 + 2= 54 9 + 14= 63 + 14= 49

Polynomials: Combining "Like Terms" (page 2 of 2)Sections: Polynomial basics, Combining "like terms"

Probably the most common thing you will be doing with polynomials is "combining like terms".This is the process of adding together whatever terms you can, but not overdoing it by trying toadd together terms that can't actually be combined. Terms can be combined ONLY IF they havethe exact same variable part. Here is a rundown of what's what:

4x and 3 NOT like terms The second term has no variable

4x and 3y NOT like termsThe second term now has a variable,but it doesn't match the variable ofthe first term

4x and 3x2 NOT like terms The second term now has the samevariable, but the degree is different

4x and 3x LIKE TERMS Now the variables match and thedegrees match

Once you have determined that two terms are indeed "like" terms and can indeed therefore becombined, you can then deal with them in a manner similar to what you did in grammar school.When you were first learning to add, you would do "five apples and six apples is eleven apples".You have since learned that, as they say, "you can't add apples and oranges". That is, "fiveapples and six oranges" is just a big pile of fruit; it isn't something like "eleven applanges".Combining like terms works much the same way.
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Simplify 3x + 4xThese are like terms since they have the same variable part, so I can combine the terms: three

x's and fourx's makes seven x's: Copyright Elizabeth Stapel 2006-2008 All Rights Reserved3x + 4x = 7x

Simplify 2x2 + 3x 4 x2 + x + 9It is often best to group like terms together first, and then simplify:

2x2 + 3x 4 x2 + x + 9= (2x2 x2) + (3x + x) + (4 + 9)= x2 + 4x + 5

In the second line, many students find it helpful to write in the understood coefficient of1 in frontof variable expressions with no written coefficient, as is shown in red below:

(2x2 x2) + (3x + x) + (4 + 9)= (2x2 1x2) + (3x + 1x) + (4 + 9)= 1x2 + 4x + 5= x2 + 4x + 5

It is not required that the understood 1 be written in when simplifying expressions like this, butmany students find this technique to be very helpful. Whatever method helps you consistently

complete the simplification is the method you should use.Simplify 10x3 14x2 + 3x 4x3 + 4x 610x3 14x2 + 3x 4x3 + 4x 6

= (10x3 4x3) + (14x2) + (3x + 4x) 6= 6x3 14x2 + 7x 6

Warning: When moving the terms around, remember that the terms' signs move with them. Don'tmess yourself up by leaving orphaned "plus" and "minus" signs behind.Simplify 25 (x + 3 x2)The first thing I need to do is take the negative through the parentheses:

25 (x + 3 x2)= 25 x 3 + x2= x2 x + 25 3= x2 x + 22If it helps you to keep track of the negative sign, put the understood 1 in front of the parentheses:

25 (x + 3 x2)= 25 1(x + 3 x2)= 25 1x 3 + 1x2= 1x2 1x + 25 3= 1x2 1x + 22= x2 1x + 22

While the first format (without the 1's being written in) is the more "standard" format, either formatshould be acceptable (but check with your instructor). You should use the format that works mostsuccessfully for you.

Simplify x + 2(x [3x 8] + 3)Warning: This is the kind of problem that us math teachers love to put on tests (yes, we're cruelpeople), so you should expect to need to be able to do this.This is just an order of operations problem with a variable in it. If I work carefully from the insideout, paying careful attention to my "minus" signs, then I should be fine:

x + 2(x [3x 8] + 3)= x + 2(x 1[3x 8] + 3)= x + 2(x 3x + 8 + 3)= x + 2(2x + 11)
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= x 4x + 22= 3x + 22

Simplify [(6x 8) 2x] [(12x 7) (4x 5)] I'll work from the inside out:

[(6x 8) 2x] [(12x 7) (4x 5)]= [6x 8 2x] [12x 7 4x + 5]

= [4x 8] [8x 2]= 4x 8 8x + 2= 4x 6

Simplify4y [3x + (3y 2x + {2y 7} ) 4x + 5]4y [3x + (3y 2x + {2y 7} ) - 4x + 5]

= 4y [3x + (3y 2x + 2y 7) - 4x + 5]= 4y [3x + (2x + 5y 7) 4x + 5]= 4y [3x 2x + 5y 7 4x + 5]= 4y [3x 2x 4x + 5y 7 + 5]= 4y [3x + 5y 2]= 4y + 3x 5y + 2= 3x 4y 5y + 2= 3x 9y + 2

If you think you need more practice with this last type of problem (with all the brackets and thenegatives and the parentheses, then review the "Simplifying with Parentheses" lesson.)

Warning: Don't get careless and confuse multiplication and addition. This may sound like a sillything to say, but it is the most commonly-made mistake (after messing up the order ofoperations):

(x)(x) = x2 (multiplication)x + x = 2x (addition)" x2 " DOES NOT EQUAL " 2x "

So if you have something like x3 + x2, DO NOT try to say that this somehow equals something

like x5

or5x. If you have something like 2x + x, DO NOT say that this somehow equalssomething like 2x2.There are two cases for dividing polynomials: either the "division" is really just a simplification andyou're just reducting a fraction, or else you need to do long polynomial division (which is coveredon the next page).

SimplifyThis is just a simplification problem, because there is only one term in the polynomial that you'redividing by. And, in this case, there is a common factor in the numerator (top) and denominator(bottom), so it's easy to reduce this fraction. There are two ways of proceeding. I can split thedivision into two fractions, each with only one term on top, and then reduce:

...or else I can factor out the common factor from the top and bottom, and then cancel off:

Either way, the answer is the same: x + 2

SimplifyAgain, I can solve this in either of two ways: by splitting up the sum and simplifying each fractionseparately: Copyright Elizabeth Stapel 1999-2009 All Rights Reserved
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