final project yes

8/7/2019 Final Project yes

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POLYNOMIALS

Polynomial is an algebraic expression that is a sum of terms contains only variables with whole numberexponents and integer coefficients. It also contains four fundamental operations applied on polynomials.

Definitions of Basic Terms in Polynomials

1. A constant is a symbol that assumes one specific value.

2. A variable is a symbol that assumes many values.

3. An algebraic expression, or simply an expression, is a collection of constants and variables involvingatleast one of the basic operations in mathematics.

4. A term is an expression preceeded by plus or minus sign.

Example:

One Term: r2

; (a + b)4

Two Terms: ab- 3; 2-r2

5. A monomial is a term involving only the product of a real number and variables with non-negativeintegral exponents.

Example:

Monomial: 6, 3b, 15xyz2, x2y4

Not Monomial: x+2 ; x/y ;

6. A polynomial is a sum of finite numbers of monomials. The general polynomial in one variable ofdegree n is of the form

Anxn + + a1x+a0

A binomial is a polynomial consisting of the sum of two monomials. Example: x2 x.

A trinomial is a polynomial consisting the sum of 3 monomials. Example: x3 x2 + 7.

7. The degree of polynomial in x is the greatest exponent occurring in the variable x.

For example x4- x 7and 2-x3 x4 have the degrees 7 and 4, respectively. A constant has 0 degree. Thedegree of (2x)0 is 0, while x20 is 1. The expressions x and 1/x have no degree since they are notpolynomials.

8. If a monomial is expressed as a product of two or more symbols, each of the symbols is called thecoefficient of the rest of the product. In 2xy, 2 is called a numerical coefficient, and xy is called the literalcoefficient.


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9. Two monomials (or two terms) are similar if they have the same literal coefficient. Example: 2x and-3x are similar terms; -3x and 4xy are dissimilar terms.

ADDITION OF EXPRESSIONS AND POLYNOMIALS

Rule 1: To add two or more monomials with the same literal coefficient, add only their numerical

coefficients and affix the literal coefficient. For Example:

A. -8x +15x = (-8+15) x = 7x

B. -8y ( -15y) = (-8 +15)y = 7y

Rule 2: To add two or more polynomials, add similar or like terms together.

Example:

A. 3x2

4x -4y, 7x2

2y 2 and -4x2

+ x y 7

B. 8x2 7x 2y, 5x2 6x 15y, and -4x2 + 11x 9y

Solution: We write the polynomials in horizontal form and perform the addition.

3x2 4x - 4y7x2 2y 2

-4x2 + x y 7________________6x2 -3x 7y 9

Example2. a. Subtract 4x y 3 from 2x y -4.b. Subtract 4x + 3y +5 from the sum of -3x y +5 and x +8y -3c. Subtract the sum of 2x 9y 8 and 6x +4y from 2x -5y 7d. Subtract the sum of 12x 14y and 6y -9 from the sum of 7x-2y +3 and 4x -5y 8.Solutions:

a. 2x y- 4 2x- y 4 b. -3x y + 5

- (4x y 3) -4x + y + 3 + x +8y -3

-2x -1 -2x +7y +2 -2x +7y +2

-(4x +3y +5)-4x -3y -5 (+)-6x +4y -3

SYMBOLS OF GROUPING EXPRESSIONS

Addition of algebraic expressions frequently involves the symbols of grouping such as parentheses (),Brackets [], and braces {}.

Rule 3. To move a grouping symbol preceded by a

i) minus sign, change the sign of each terms;


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ii.) plus sign, no further change is done;

iii.) factor, use the distributive law.

Example: Perform the indicated operations

a. (2x y +10) + (4x 3y) 2(3x 4y +6)

b. 4x 2y 5 - 2(8x - 7y) (x 4y 1)

Solution:

(i) (ii) (iii)

(2x y +10) + (4x 3y) 2(3x 4y +6)

= -2x + y - 10 + 4x 3y - 6x + 8y 12

= -2x +4x -6x +y -3y +8y 10 -12 = -4x+ 6y 22

Rule 4. When one symbol of grouping is within another symbol of grouping the innermost symbol mustbe removed first.

Example: Simplify the following

a. {-2 y [3x (4x + y 3) y] 7}

b. {-3x +4y (7x 8)} {3x [-4y - (x 5)]}

c. {9x 6 [3x (4x 5y 7)] 8y}

Solutions:

a. {-2 y [3x (4x + y 3) y] 7} b. {-3x +4y (7x 8)} {3x [-4y - (x 5)]}

= - {- 2x y [3x 4x y + 3 y] 7} = {-3x +4y (7x 8)} {3x [-4y - (x -5)]}

= - {-2x y [-x 2y +3] -7} = - {-4x +4y + 8} {3x +4y + x -5}

= - {-2x-y +x +2y 3 -7} = 4x 4y 8 {4x + 4y 5}

= - {-x +y -10} = 4x 4y 8 4x 4y + 5

= x-y +10 = -8y - 3


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Value of Polynomial

The value of Polynomial or expression in x is obtained by substituting a certain given value forx.

Example, the value of 2x3

4x2

x 5 at x is = -1

2(-1)3 4(-1)2 (-1) 5 = -10

The value of Polynomial or Expression at two limits denoted by f (x) |x=bx =a is defined as the differenceof the values of f (x) at x + b and that of f (x) at x = a. The number b is called the upper limit ofthe polynomial or expression, while a is the lower limit. For Example:

[-(2)2 -4(2) 1] [-(-1)2 4(-1) 1] = -15

Also, the value of x-1/ 2x + 5 |10 is 0 - (-1/5) = 1/5; x-a /2x + 5 | 2-1 is 1/9 (2/3) = 7/9

POWERS WITH THE POSITIVE INTEGRAL EXPONENTS

A compact notation for the product of n factors each of which is a is a given in the followingdefinition.

The power an is defined as follows:

an = a a a . . . a, n is a positive integer|_____ n factors____|

We call the exponent n the exponent of a as the base.

The power is one strong concept in mathematics to put a number or expression in a morecompact form. Some notations employed in this concept which convey different notations suchas the following

-32 = -3 3 = 9 ; (-3)2 = (-3)(-3) = 9; -(3)2 = -(3 3) = -9

(-2)3 = (-2) (-2) (-2) = -8 ; -54 = -5 5 5 5 = -625 ; -23 = -2 2 2 = 8

LAWS OF EXPONENTS1. The product of powers.

am an = am+ n

Illustrations: 1. x2 . x3 = x5 2. x4 x5 = x9 3. (2x3)(-3x4) = -6x7


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2. The quotient of powers.

am

an = am-n

Illustrations: 1. (x2)3 = x 6 2. (4x3)2 = 16x6 3. (3x4c) 5 = 243x2() c5

3. The power of product.

(ab)m = ambm

Illustrations: 1. (2x)3 = 23 x3 = 8x3 2. -2(x2)3 = -2x6

4. The power of power.

(am )n = amn

5. The power of quotient.

(a / b)m = am / bm

Illustration: (-2x3/3)4 = (-2)4(x3)4/ 34 = 16x12/81

THE PRODUCTS OF POLYNOMIAL

Rule1. To multiply two monomials, use commutative, associative and the laws of exponents inmultiplication. Example:

(3x2y4z2w9) (-4xyz4wv)

= 3 (-4) (x2x) (y4y) (z2z4) (w9w) v

= -12 x10 y6 w10 v

Rule 2. To multiply two polynomials, use the distributive law and apply rule 1.

Example:

(2x 3y) (4x + 5y)

= 2x (4x +5y) 3y (4x + 5y)

= 8x2 2xy 15y2


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DIVISION OF POLYNOMIALS

Rule 1. To divide monomials, use the laws of exponents in division.

a. -6x4y6zw = -6 . x4 . y6 . z . w = x3w12xy8 zu 12 x y8 z u 2y2 u

Rule 2. To divide polynomial by a monomial, we use : a+b = a + bc c + c

The preceding theorem is applied to a finite number of terms in a given polynomial.

Example: (4x5+18x2+16x) (2x)= (4x5 2x) + (18x22x) + (16x 2x)= (2x4) + (9x) + (8)= 2x4+ 9x +8

Rule 3. The last rule is to divide a polynomial by another polynomial with at least two terms. This type ofdivisions is applied only when the degree of the polynomial in the numerator is greater or equal to thedegree of the polynomial in the denominator.

1. Arrange the terms of the dividend in descending powers of the variable.

2. Divide the first term in the dividend by the first term of the divisior, giving the first term of thequotient.

3. Multiply each term of the divisor by the first term of the quotient and subtract the product of thedividend.

4. Use the remainder obtained in step 3 as a new dividend, and repeat steps 2 and 3.

5. Continue the process until the remainder is reached whose degree should be less than the degree of thedivisor.

The result of division is expressed as follow:

a. for exact division (remainder 0) b. for remainder 0

dividend = quotient dividend = quotient(Q) + remainder (R )divisor divisor divisor (D)

Example : x2 + 2x 3 = x +3 x2 + 2x 3 = x+ 4 + _5_x-1 x-2 x -2

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


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SYNTHETIC DIVISION

Another method of division which has a very short and simple procedure is called Synthetic Division.Unlike the usual division which involves the four fundamental operations, this method requires onlyaddition and multiplication applied to the coefficients. This method is applied when the divisor is of theform x + a.

Steps to in Synthetic Divison

1. Arrange the terms of the dividend in descending powers of variable.

2. Write the numerical coefficients of each term of the dividend in a row indicating the coefficients ofpowers. Replace the missing power with the zero coefficient.

3. Replace the divisor x r by r: for divisor x + r, replace it with r (constant divisor).

4. Multiply the coefficient of the largest power of x, written on the third row, by the constant divisor.Place the product beneath the coefficient. Multiply the sum by the constant divisor and place it beneaththe coefficient of the next largest power. Continue this procedure until there is a product added to theconstant of the last term.

5. The last number on the third tow is called the remainder, the rest of the numbers, starting from the leftto right, are the coefficients of the terms in the quotient, which is one degree less than that of thedividend.

Example:

f(x) = 4x3 - 3x2 + x - 4

We will picture it evaluated at the input value x = 2. Arrange the input value, the coefficients, and a linelike this:

2) 4 -3 1 -4

---------------

Now drop down the 4:

2) 4 -3 1 -4

---------------4

Multiply the input 2 times the 4. Place this product, 8, under the -3:

2) 4 -3 1 -4


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8---------------4

Add the -3 and the 8. Place this sum, 5, under the line:

2) 4 -3 1 -48

---------------4 5

Multiply the input 2 times the 5. Place this product, 10, under the 1:

2) 4 -3 1 -48 10

---------------4 5

Add the 1 and the 10. Place this sum, 11, under the line:

2) 4 -3 1 -48 10

---------------

4 5 11

Multiply the input 2 times the 11. Place this product, 22, under the -4:

2) 4 -3 1 -48 10 22

---------------4 5 11

Add the -4 and the 22. Place this sum, 18, under the line:

2) 4 -3 1 -48 10 22

---------------4 5 11 18

Thus, the answer is 4x2 + 5x + 11.


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FACTORING SPECIAL PRODUCTS

A. Factoring using the GCF

1. Find the largest number common to every coefficient or number.

2. Find the GCF of each variable. It will always be the variable raised to the smallest exponent.

3. Find the terms that the GCF would be multiplied by to equal the original polynomial.

It looks like the distributive property when in factored form .. GCF(terms).

Examples:

3x2 - 6x

2x2 - 4x + 8

5x2y3 + 10x3y

B. Factoring the Difference of Two Squares

1. The factors will always be (a + b)(a b).

2. "a" is the square root of the first term

3. "b" is the square root of the second term.

Examples:

9x2 - 49

121x2 - 100

25x2 - 64y2

C. Factoring a Perfect Square Trinomial

1. Characteristics

"ax2

" term is a perfect square."c" term is a perfect square.

"c" term is positive.

Factors into two identical binomials: (a b)2.


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2. Steps to Factor

Write it as (a b)2 since it factors into two identical binomials.

"a" is the square root of the "ax2" term.

"b" is the square root of the "c" terml.

The operation in the binomial factor is the same as the operation in front of the "x" term.

Examples:

9x2 - 30x + 25

4x2 + 28xy + 49y2

2x2 + 16x + 32

16x

3

+ 80x

2

+ 100x


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RATIONAL EXPRESSIONS

A rational expression is one that can be written in the form

where Pand Q are polynomials and Q does not equal 0.

An example of a rational expression is:

Domain of a Rational Expression

With rational functions, we need to watch out for values that cause our denominator to be 0. Ifour denominator is 0, then we have an undefined value.

So, when looking for the domain of a given rational function, we use a back doorapproach. We find the values that we cannot use, which would be values that make thedenominator 0.

Example 1: Find all numbers that must be excluded from the domain of .

Our restriction is that the denominator of a fraction can never be equal to 0.

So to find what values we need to exclude, think of what value(s) ofx, if any, wouldcause the denominator to be 0.

*Factor the den.

This give us a better look at it.

Since 1 would make the first factor in the denominator 0, then 1 would have to beexcluded.

Since - 4 would make the second factor in the denominator 0, then - 4 would also


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have to be excluded.

Fundamental Principle of Rational Expressions

For any rational expression , and any polynomial R, where , , then

In other words, if you multiply the EXACT SAME thing to the numerator and denominator,then you have an equivalent rational expression.

This will come in handy when we simplify rational expressions, which is coming up next.

Simplifying (or reducing) a Rational Expression

Step 1: Factor the numerator and the denominator.

Step 2: Divide out all common factors that the numerator and the denominator have.

Example 2: Simplify and find all numbers that must be excluded from the domain of the simplified

rational expression: .

Step 1: Factor the numerator and the denominator

AND



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*Factor the trinomials in the num. and den.

*Divide out the common factor of (x + 3)

*Rational expression simplified

To find the value(s) needed to be excluded from the domain, we need to askourselves, what value(s) ofx would cause our denominator to be 0?

Looking at the denominatorx - 9, I would say it would have to be x = 9. Dont youagree?

9 would be our excluded value.

Example 3: Simplify and find all numbers that must be excluded from the domain of the

simplified rational expression:

Step 1: Factor the numerator and the denominator

AND

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*Factor the diff. of squares in the num. and*Factor the trinomial in the den.

*Factor out a -1 from (5 - x)

*Divide out the common factor of (x - 5)

*Rational expression simplified

Note that 5 - x and x - 5 only differ by signs, in other words they are opposites of each other. In that case,you can factor a -1 out of one of those factors and rewrite it with opposite signs, as shown in line 3above.

To find the value(s) needed to be excluded from the domain, we need to ask ourselves, what value(s)ofx would cause our denominator to be 0?

Looking at the denominatorx - 5, I would say it would have to be x = 5. Dont you agree?

5 would be our excluded value.

Multiplying Rational Expressions

Q and S do not equal 0.

Step 1: Factor both the numerator and the denominator.

Step 2: Write as one fraction.Write it as a product of the factors of the numerators over the product of the factors of the denominators.DO NOT multiply anything out at this point.Step 3: Simplify the rational expression.

Step 4: Multiply any remaining factors in the numerator and/or denominator.

Example 1: Multiply .
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#diffsqhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#tri1http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#diffsqhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#tri1


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Step 1: Factor both the numerator and the denominator

AND

Step 2: Write as one fraction.

*Factorthe num. and den.

In the numerator we factored a difference of squares.

In the denominator we factored a GCF and a trinomial.

Step 3: Simplify the rational expression.

AND


*Simplifyby div. out the common factors of (y + 3), (y -3) and y

*Excluded values of the original den.

Note that even though all of the factors in the numerator were divided out there isstill a 1 in there. It is easy to think there there is "nothing" left and make thenumerator disappear. But when you divide a factor by itself there is actually a 1there. Just like 2/2 = 1 or 5/5 = 1.

Also note that the values that would be excluded from the domain are 0, 3, -6, and-3. Those are the values that makes the original denominator equal to 0.
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Example 2: Multiply .

Step 1: Factor both the numerator and the denominator

AND

Step 2: Write as one fraction.


In the numerator we factored adifference of cubes and a GCF.

In the denominator we factored a trinomial.

Step 3: Simplify the rational expression.

AND


*Simplifyby div. out the common factors of(x - 3), 2, and (x + 2)


Note that the values that would be excluded from the domain are 0, 3, and -2. Those arethe values that makes the original denominator equal to 0.
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Dividing Rational Expressions

where Q, S, and R do not equal 0.

Step 1: Write as multiplication of the reciprocal.

Step 2:Multiply the rational expressions as shown above.

Example 3: Divide .

Step 1: Write as multiplication of the reciprocal

AND

Step 2: Multiply the rational expressions as shown above.

*Rewrite as mult. of reciprocal


*Simplifyby div. out the common factors of3x and (x + 6)

*Multiplythe den. out

*Excluded values of the original den. ofproduct

In the numerator of the product we factored a GCF.

In the denominator we factored a trinomial.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#gcfhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#tri1http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#gcfhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htm#tri1


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Note that the values that would be excluded from the domain are -6 and 0.Those are the values that makes the original denominator of the productequal to 0.

Example 4: Divide .

Step 1: Write as multiplication of the reciprocal

AND

Step 2: Multiply the rational expressions as shown

above.

*Rewrite as mult. ofreciprocal


*Simplifyby div. out thecommon factors ofy, (y + 4), and (y - 4)

*Multiply the num. and den.out

*Excluded values of theoriginal den. of quotient &product

In the numerator of the product we factored a GCF and a trinomial.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut9_mulrat.htm#multiplyhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut7_factor.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut8_simrat.htmhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut6_poly.htm


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In the denominator we factored a GCF and a difference of squares.Note that the values that would be excluded from the domain are 0, 2, - 4, 4, and -3. Those are the valuesthat make the original denominator of the quotient and the product equal to 0.

Adding or Subtracting Rational Expressions

with Common Denominators

Step 1: Combine the numerators together.

Step 2: Put the sum or difference found in step 1 over the common denominator.

Step 3: Reduce to lowest terms as shown in Tutorial 8: Simplifying Rational Expressions.

Why do we have to have a common denominator when we add or subtract rationalexpressions?????

Good question. The denominator indicates what type of fraction that you have and the numerator iscounting up how many of that type you have. You can only directly combine fractions that are of thesame type (have the same denominator). For example if 2 was my denominator, I would be counting uphow many halves I had. If 3 was my denominator, I would be counting up how many thirds I had. But Iwould not be able to add a fraction with a denominator of 2 directly with a fraction that had a

denominator of 3 because they are not the same type of fraction. I would have to find a commondenominator first, which we will cover after the next two examples.

Example 1: Add .

Since the two denominators are the same, we can go right into adding these two rational expressions.

Step 1: Combine the numerators together

AND

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*Common denominator of 5x - 2

*Combine the numerators

*Write over common denominator


Step 3: Reduce to lowest terms.

Note that neither the numerator nor the denominator will factor. The rational expression is as simplifiedas it gets.

Also note that the value that would be excluded from the domain is 2/5. This is the value that makes theoriginal denominator equal to 0.

Example 2: Subtract

Since the two denominators are the same, we can go right intosubtracting these two rational expressions.

Step 1: Combine the numerators together

AND


*Common denominator ofy - 1

*Combine the numerators*Write over common denominator

Step 3:Reduce to lowest terms.
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*Factor the num.

*Simplify by div. out the common factor of (y - 1)


Note that the value that would be excluded from the domain is 1. This is the value that makes the original

denominator equal to 0.Least Common Denominator (LCD)

Step 1: Factor all the denominators

Step 2: The LCD is the list of all the DIFFERENT factors in the denominators raised to the highest powerthat there is of each factor.

Adding and Subtracting Rational Expressions Without a Common Denominator

Step 1:Find the LCD as shown above if needed.

Step 2: Write equivalent fractions using the LCD if needed.If we multiply the numerator and denominator by the exact same expression it is the same as multiplyingit by the number 1. If that is the case, we will have equivalent expressions when we do this.

Now the question is WHAT do we multiply top and bottom by to get what we want? We need to have theLCD, so you look to see what factor(s) are missing from the original denominator that is in the LCD. Ifthere are any missing factors then that is what you need to multiply the numerator AND denominator by.

Step 3: Combine the rational expressions as shown above.

Step 4: Reduce to lowest terms as shown in Tutorial 8: Simplifying Rational Expressions.

Example 3: Add .

Step 1: Find the LCD as shown above if needed.

The first denominator has the following two factors:
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*Factor theGCF

The second denominator has the following factor:

Putting all the different factors together and using the highest exponent, we getthe following LCD:

Step 2: Write equivalent fractions using the LCD if needed.

Since the first rational expression already has the LCD, we do not need to change this fraction.

*Rewriting denominator in factored form

Rewriting the second expression with the LCD:

*Missing the factor of (y - 4) in the den.

*Mult. top and bottom by (y - 4)

Step 3:Combine the rational expressions as shown above.
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*Simplify by div. out the common factor ofy


Note that the values that would be excluded from the domain are 0 and 4. These are the values that makethe original denominator equal to 0.

Example 4: Add .

Step 1: Find the LCD as shown above if needed.The first denominator has the following factor:

The second denominator has the following two factors:

*Factor thedifference of squares

Putting all the different factors together and using the highest exponent, we get the following LCD:

Step 2: Write equivalent fractions using the LCD if needed.Rewriting the first expression with the LCD:
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*Missing the factor of (x+ 1) in the den.*Mult. top and bottom by (x+ 1)

Since the second rational expression already has the LCD, we do not need to change this fraction

*Rewriting denominator in factored form

Step 3: Combine the rational expressions as shown above.



Step 4: Reduce to lowest terms.

This rational expression cannot be simplified down any farther.

Also note that the values that would be excluded from the domain are -1 and 1. These are the valuesthat make the original denominator equal to 0.

Example 5: Subtract .

Step 1: Find the LCD as shown above if needed.

The first denominator has the following two factors:
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*Factor the trinomial

The second denominator has the following two factors:

*Factor the trinomial

Putting all the different factors together and using the highest exponent, we get the following LCD:

Step 2: Writeequivalent fractionsusing the LCD if needed.

Rewriting the first expression with the LCD:

*Missing the factor of (x - 8) in theden.*Mult. top and bottom by (x - 8)

Rewriting the second expression with the LCD:

*Missing the factor of (x + 5) in theden.*Mult. top and bottom by (x + 5)

Step 3:Combine the rational expressions as shown above.
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*Distribute the minus sign throughthe ( )


*Factor the num.

*No common factors to divide out


Note that the values that would be excluded from the domain are -5, -1 and 8. These are the values thatmake the original denominator equal to 0.
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LOGARITHM

The logarithm for a base and a number is defined to be theinverse functionof takingto the power , i.e., . Therefore, for any and ,

or equivalently,

For any base, the logarithm function has a singularityat . In the above plot, the blue curve is

the logarithm to base 2 ( ), the black curve is the logarithm to base (thenatural

logarithm ), and the red curve is the logarithm tobase 10 (thecommon logarithm,

i.e., log, ).

Note that while logarithm base 10 is denoted in this work, on calculators, and in elementary

algebra and calculus textbooks, mathematicians and advanced mathematics texts uniformly use
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the notation to mean , and therefore use to mean thecommon logarithm. Extreme

care is therefore needed when consulting the literature.

The situation is complicated even more by the fact that number theorists (e.g., Ivi 2003)

commonly use the notation to denote the nested natural logarithm .

In Mathematica, the logarithm to the base is implemented as Log[b, x], while Log[x] gives

the natural logarithm, i.e., Log[E, x], where E is the Mathematica symbol fore.

Whereas powers of trigonometric functions are denoted using notations like , is less

commonly used in favor of the notation .

Logarithms are used in many areas of science and engineering in which quantities vary over a

large range. For example, the decibel scale for the loudness of sound, the Richter scale of

earthquake magnitudes, and the astronomical scale of stellar brightnesses are all logarithmic

scales.

The derivative and indefinite integral of are given by

The logarithm can also be defined forcomplex arguments, as shown above. If the logarithm is

taken as the forward function, the function taking the baseto a givenpoweris then called

the antilogarithm.
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For , is called the characteristic, and is called the mantissa.

Division and multiplication identities for the logarithm can be derived from the identity

including

There are a number of properties which can be used to change from one logarithmbase to

another, including
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An interesting property of logarithms follows from looking for a number such that

so

Another related identity that holds for arbitrary is given by

Numbers of the form are irrationalif and are integers, one of which has a prime factor

which the other lacks. A. Baker made a major step forward intranscendental numbertheory by

proving the transcendence of sums of numbers of the form for and algebraic numbers.
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RADICALS

WE SAY THAT A SQUARE ROOT RADICAL is simplified, or in its simplest form, when

the radicandhas no square factors.

A radical is also in simplest form when the radicand isnot a fraction.

Example 1. 33, for example, has no square factors. Its factors are 3 11, neither of which is

a square number. Therefore, 33 is in its simplest form.

Example 2. 18 has the square factor 9. 18 = 9 2. Therefore, 18 is not in its simplest

form. To put a radical in its simplest form, we make use of this theorem:

ab=ab

The square root of a product

is equal to the product of the square roots

of each factor.

(We will prove that when we come to rational exponents, Lesson 29.

Here is a simple illustration: 100=425=425=25=10

Therefore,

18= 92= 92= 32

We have simplified 18
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Example 3. Simplify .

Solution. We have to factor 42 and see if it has any square factors. We can begin the

factoring in any way. For example,

42 = 6 7

We can continue to factor 6 as 2 3, but we cannot continue to factor 7, because 7 is

aprime number(Lesson 31 of Arithmetic). Therefore,

42 = 2 3 7

We now see that 42 has no square factors -- because no factor is repeated. Compare

Example 1 and Problem 2 of theprevious Lesson.

therefore is in its simplest form.

Example 4. Simplify .

Solution.

180 = 2 90 = 2 2 45 = 2 2 9 5 = 2 2 3 3 5

Therefore,

= 2 3 = 6 .

Problem 1. Simplify the following. Inspect each radicand for a square factor: 4, 9, 16, 25,

and so on.
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To see the answer, pass your mouse over the colored area.

To cover the answer again, click "Refresh" ("Reload").

Do the problem yourself first!

a) =

b) = = = 5

c) = = = 3

d) = = 7

e) = = 4

f) = = 10

g) = = 5

h) = = 4

Problem 2. Reduce to lowest terms.

a)2

=2

=2

=

b)3

=3

=3

= 2

c) = The radical is in its simplest form. The fraction cannot be reduced.


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2

Similar radicals

Similar radicals have the same radicand. We add them as like terms.

7 + 2 + 5 + 6 = 7 + 8 + 4 .

2 and 6 are similar, as are 5 and . We combine them by adding their

coefficients.

As for 7, it does not "belong" to any radical.

Problem 3. Simplify each radical, then add the similar radicals.

a) + = 3 + 2 = 5

b) 4 2 + = 4 2 +

= 4 5 2 7 +

= 20 14 +

= 7


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c) 3 + 2 = 3 + 2

= 3 2 + 2 2 4

= 6 + 2 8

= 2 2

d) 3 + + = 3 + +

= 3 + 2 + 3

= 3 + 5

e) 1 + = 1 +

= 1 8 + 3

= 1 5

Problem 4. Simplify the following.


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a)2

=2

= 2 ,on dividing each term in the numerator by 2.

To see that 2 is a factor of the radical, we first have to simplify the radical. SeeProblem 2.

b)5

=5

= 2 +

c)6

=6

=

3on dividing each termby 2.

A logarithm is just an exponent.

To be specific, the logarithm of a numberx to a base b is just the exponent you put onto b to

make the result equal x. For instance, since 5 = 25, we know that 2 (the power) is the logarithm

of 25 to base 5. Symbolically, log5(25) = 2.

More generically, ifx = by, then we say that y is the logarithm ofx to the base b or the

base-blogarithm ofx. In symbols, y = logb(x). Every exponential equation can be rewritten as a

logarithmic equation, and vice versa, just by interchanging the x and y in this way.
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Another way to look at it is that the logbx function is defined as the inverse of

the bx function. These two statements express that inverse relationship, showing how an

exponential equation is equivalent to a logarithmic equation:

x = by is the same as y = logbx

Example 1: 1000 = 103 is the same as 3 = log101000.

Example 2: log381 = ? is the same as 3? = 81.

It cant be said too often: a logarithm is nothing more than an exponent. You can write the above

definition compactly, and show the log as an exponent, by substituting the second equation into

the first to eliminatey:

Read that as the logarithm ofx in base b is the exponent you put on b to get x as a result.

Before pocket calculators only three decades ago, but in student years thats the age of

dinosaurs the answer was simple. You needed logs to compute most powers and roots with

fair accuracy; even multiplying and dividing most numbers were easier with logs. Every decent

algebra books had pages and pages of log tables at the back.

The invention of logs in the early 1600s fueled the scientific revolution. Back then

scientists, astronomers especially, used to spend huge amounts of time crunching numbers on
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paper. By cutting the time they spent doing arithmetic, logarithms effectively gave them a longer

productive life. The slide rule, once almost a cartoon trademark of a scientist, was nothing more

than a device built for doing various computations quickly, using logarithms. See Eli Maors e:

The Story of a Numberfor more on this.

Today, logs are no longer used in routine number crunching. But there are still good reasons

for studying them.

Why do we use logarithms, anyway?

To find thenumber of payments on a loan or the time to reach an investment goal

To model many natural processes, particularly in living systems. We perceive

loudness of sound as the logarithm of the actual sound intensity, and dB (decibels) are

a logarithmic scale. We also perceive brightness of light as the logarithm of the actual

light energy, and star magnitudes are measured on a logarithmic scale.

To measure the pH or acidity of a chemical solution. The pH is the negative

logarithm of the concentration of free hydrogen ions.

To measure earthquake intensity on the Richter scale.

To analyze exponential processes. Because the log function is the inverse of the

exponential function, we often analyze an exponential curve by means of logarithms.

Plotting a set of measured points on log-log or semi-log paper can reveal such

relationships easily. Applications include cooling of a dead body, growth of bacteria,

and decay of a radioactive isotopes. The spread of an epidemic in a population often

follows a modified logarithmic curve called a logistic.
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To solve some forms of area problems in calculus. (The area under the curve 1/x,

between x=1 and x=A, equals ln A.)

Bases

From the definition of a log as inverse of an exponential, you can immediately get some basic

facts. For instance, if you graph y=10x (or the exponential with any other positive base), you see

that its range is positive reals; therefore the domain ofy=log x (to any base) is the positive reals.

In other words, you cant take log 0 or log of a negative number.

(Actually, if youre willing to go outside the reals, you can take the log of a negative

number. The technique is taught in many trigonometry courses.

You know thatanything to the zero power is 1: b0 = 1. Change that to logarithmic form with

the definition of logsand you have

logb1 = 0 for any base b

In the same way, you know that the first power of any number is just that number: b1 = b. Again,

turn that around to logarithmic form and you have

logbb = 1 for any base b

Example 3: ln 1 = 0

Example 4: log55 = 1
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Logarithm as Inverse

A log is an exponent because the log function is the inverse of the exponential function. The

inverse function undoes the effect of the original function. (Im not a big fan of most uses the

term cancel in math, but it does fit in this situation.)

This means that if you take the log of an exponential (to the same base, of course), you get back

to where you started:

logbbx = x for any base b

This fact lets you evaluate many logarithms without a calculator.

Example 5: log5125 = log5(5) = 3

Example 6: log10103.16 = 3.16

Example 7: ln e-kt/2 = -kt/2

Any positive number is suitable as the base of logarithms, but two bases are used more than any

others:

base of

logarithmssymbol name

10log

(if no base shown)common logarithm

e ln natural logarithm,
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pronounced ell-enn or lahn

Natural logs are logs, and follow all the same rules as any other logarithm. Just remember

ln x means logex

Why base e? Whats so special about e? Most of the explanations need some calculus, for

instance that exis the only function that is both its own integral and its own derivative or that e

has this beautiful definition in terms of factorials:

e = 1/0! + 1/1! + 1/2! + 1/3! + ...

Numerically, e is about 2.7182818284. Its irrational (the decimal expansion never ends and

never repeats), and in fact like pi its transcendental (no polynomial equation with integer

coefficients has pi oreas a root.)

e (like pi) crops up in all sorts of unlikely places, like computations of compound interest. It

would take a book to explain, and fortunately there is a book, Eli Maors e: The Story of a

Number. He also goes into the history of logarithms, and the book is well worth getting from

your library.

Logarithms with same base

In a minute well look at the various combinations. But first you might want to know the general

principle:logs reduce operations by one level. Logs turn a multiplication into an addition, a

division into a subtraction, an exponent into a multiplication, and a radical into a division. Now

lets see why, and look at some examples.

Multiplying Logarithms


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Multiplying two expressions corresponds to adding their logarithms. Can we make sense of this?

By the compact definition,

x = blogbx and y = blogby

and therefore, substituting forx and y,

xy = blogbxblogby

But when youmultiply two powers of the same base, you add their exponents. So the right-hand

side becomes

xy = blogbx+logby

Now apply the compact definition to the left=hand side:

blogb(xy) = xy

Combine that with the preceding equation to obtain

blogb(xy) = blogbx+logby

Now we have two powers of the same base. If the powers are equal, then the exponents must also

be equal. Therefore

logb(xy) = logbx + logby

So whats the bottom line? Multiplying two numbers and taking the log is the same as taking

their logs and adding.

Example 8: log8(x)+log8(x) is the same as log8(xx) or just log8(x).

Example 9: log10(20)+log10(50) = log10(2050) = log10(1000) = 3.
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Exponent of a Logarithm

Continuing our theme of logarithms reducing the level of operations, if you have the yth power

of a number and take the log, the result is y times the log of the number. Heres why, starting

with xy:

Start with the compact definition of a logarithm:

x = blogbx

and raise both sides to the y power:

xy = (blogbx)y

A power of a poweris equivalent to just multiplying the exponents. Simplify the right-hand side:

xy = b(y logbx)

Rewrite the left-hand side using the compact definitionof a log:

blogb(xy) = xy

(The font may be hard to read: thats x to the powery on left and right.) and combine the last two

equations:

blogb(xy) = b(y logbx)

If the powers are equal and the bases are equal, the exponents must be equal:

logb(xy) = y logbx

Example 10: ln(26) = 6 ln 2 (where ln means loge, thenatural logarithm).
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Example 11: log5(5x) is notequal to 2 log5(5x). Be careful with order of operations! 5x is 5(x),

not (5x). log5(5x) must first be decomposed as the log of the product: log55 + log5(x). Then the

second term can use the power rule, log5(x) = 2 log5x. The first term is just 1. Summing up,

log5(5x) = 1 + 2 log5x.

Division of Logarithms

Since division is the opposite of multiplication, and subtraction is the opposite of addition, its

not surprising that dividing two expressions corresponds to subtracting their logs. While we

could go back again to thecompact definition, its probably easier to use the two preceding

properties.

Start with the fact that 1/y = y1 (see the definition ofnegative exponents):

x/y = x(1/y) = xy1

and take the log of both sides:

logb(x/y) = logb(xy1)

The right-hand side is the log of a product, which becomes the sum of the logs:

logb(x/y) = logbx + logb(y1)

and the second term is the log of a power, which becomes (1) times the log, or just minus the

log:

logb(x/y) = logbx logby

In words, if you divide and take the log, thats the same as subtracting the individual logs.
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Example 12: 67515=45, and therefore log10675 log1015 = log1045. (Try it on your calculator!)

Example 13: log(xy) log(xy) = log(xy / xy) = log(x/y) = log(x) log(y).

Changing the Base

Now you have everything you need to change logarithms from one base to another. Look again

at the compact equation that defines a log in base b:

To change the log from base b to another base (call it a), you want to find loga(x). Since you

already havex on one side of the above equation, it seems like a good start is to take the base-

a log of both sides:

loga(blogbx) = logax

But the left-hand side of that equation is just the log of a power. You remember that log(xy) is

just log(x) times y. So the equation simplifies to

(logab) (logbx) = logax

Notice that logab is a constant. This means that the logs of all numbers in a given base a are

proportional to the logs of the same numbers in another base b, and the proportionality constant

logab is the log of one base in the other base. If youre like me, you may have trouble

remembering whether to multiply or divide. If so, just derive the equation as you see, it takes

only two steps.

Some textbooks present the change-of-base formula as a fraction. To get the fraction from the
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above equation, simply divide by the proportionality constant logab:

logbx = (logax) / (logab)

Example 14: log416 = (log 16) / (log 4). (You can verify this with your calculator, since you

know log416 must equal 2.)

Example 15: Most calculators cant graph y = log3x directly. But you can change the base

to e and easily plot y = (ln x)(ln 3). (You could equally well use base 10.)

An interesting side road leads from the above formula. Replace x everywhere with a this is

legal since the formula is true for all positive a, b, and x. You get

logba = (logaa) / (logab)

But logaa = 1 (see Log of 1 above), so the formula becomes

logba = 1 / (logab)

Example 16: log10e = 1/(ln 10). (You can verify this with your calculator.)

Example 17: log1255 = 1/(log5125). This is easy to verify: 53 = 125, and 5 is the cube root of 125.

Therefore log1255 = 1/3 and log5125 = 3, and 1/3 does indeed equal 1/3.

The laws of logarithms have been scattered through this longish page, so it might be helpful to

collect them in one place. To make this even more amazingly helpful , the associated laws

of exponents are shown here too.
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For heavens sake, dont try to memorize this table! Just use it to jog your memory as

needed. Better yet, since a log is an exponent, use the laws of exponents to re-derive any

property of logarithms that you may have forgotten. That way youll truly gain mastery of this

material, and youll feel confident about the operations.

exponents logarithms

(All laws apply for any positive a, b, x, and y.)

x = by is the same as y = logbx

b0 = 1 logb1 = 0

b1 = b logbb = 1

b(logbx) = x logbbx = x

bxby = bx+y logb(xy) = logbx + logby

bxby = bxy logb(x/y) = logbx logby

(bx)y = bxy logb(xy) = y logbx

(logab) (logbx) = logax

logbx = (logax) / (logab)

logba = 1 / (logab)

.
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Example 18: log (5+x) is not the same as log 5 + log x. As you know, log 5 + log x = log(5x),

not log(5+x). Look carefully at the above table and youll see that theres nothing you can do to

split up log(x+y) or log(xy).

Example 19: (log x) / (log y) is not the same as log(x/y). In fact, when you divide two logs to the

same base, youre working the change-of-base formula backward. Though its not often useful,

(log x) / (log y) = logyx. Just dont write log(x/y)!

Example 20: (log 5)(log x) is not the same as log(5x). You know that log(5x) is log 5 + log x.

Theres really not much you can do with the product of two logs when they have the same base.

The general rule is that logs simply drop an operation down one level: exponents become

multipliers, divisions become subtractions, and so on. If ever youre unsure of an operation,

like how to change base, work it out by using the definition of a log and applying the laws of

exponents, and you wont go wrong.

A radical is a symbol for the indicated root of a number, for example a square root or cube root;

the termis also synonymous for the root itself.

The word radical has both Latin and Greek origins. From Latin raidix, radicis means "root" and

in Greekraidix is the analog word for "branch." The concept of a radicalthe root of a number

can best be understood by first tackling the idea of exponentiation, or raising a number to a

given power. We indicate a number raised to the nth power by writing xn. This expression

indicates that we are multiplying x by itselfn number of times. For example, 32 = 3 3 = 9, and

24 = 2 2 2 2 = 16.
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Just as division is the inverse ofmultiplication, taking the root of a number is the inverse of

raising a number to a power. For example, if we are seeking the square root ofx2, which

equals x x, then we are seeking the variable that, when multiplied by itself, is equal to x2

namely, x. That is to say, 9 = 32 = 3 3. Similarly, if we are looking for the fourth root ofx 4,

then we are looking for the variable that multiplied by itself four times equals x. For example,

[fourth root of 16] x = 24 = 2 2 2 2.

The radical &NA; is the symbol that calls for the root operation; the number or variable under

the radical sign is called the radicand. It is common parlance to speak of the radicand as being

"under the radical." It is also common to simply use the term "radical" to indicate the root itself,

as when we speak of solving algebraic equations by radicals.

The expression &NA; = Pis called the radical expression, where n is the indicated root

index, R is a real number and Pis the nth root of numberR such that Pn = R.

The most commonly encountered radicals are the square root and the cube root. We have already

discussed the square root. A bare radical sign with no indicated root index shown is understood

to indicate the square root.

The cube root is the numberPthat solves the equation Pn = R. For example, the cube root of 8, is

2.Both the radicand R and the order of the root n have an effect on the root(s) P. For example,

because a negative number multiplied by a negative number is a positive number, the even

roots (n = 2, 4, 6, 8...) of a positive number are both negative and positive: 9 = 3, &NA; = 2.
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Because the root Pof &NA; must be multiplied an odd number of times to generate the

radicand R, it should be clear that the odd roots (n = 3, 5, 7, 9...) of a positive number are

positive, and the odd roots of a negative number are negative. For example, &NA; = 2 (23 = 2

2 2 = 4 2 = 8), but &NA; = 2 (23 = 2 2 2 = 4 2 = 8).

Taking an even root of a negative number is a trickier business altogether. As discussed above,

the product of an even number of negative values is a positive number. The even root of a

negative number is imaginary. That is, we define the imaginary unit i = 1 or 2 i = 1. Then

9 = 9 1 = 3i. The imaginary unit is a very useful concept in certain types

ofcalculus and complex analysis.

Multiplication of radicals

The product of two radicals with same index n can be found by multiplying the radicands and

placing the result under the same radical. For example, 9 25 = (925) = 225 = 15, which

is equal to 3 5 = 9 25. Similarly, radicals with the same index sign can be divided by

placing the quotient of the radicands under the same radical, then taking the appropriate root.

The radical of a radical can be calculated by multiplying the indexes, and placing the radicand

under the appropriate radical sign. For instance, &NA; = &NA; = 2.
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final project yes

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