c hapter 4 polynomial and rational functions. s ection 4.1 polynomial functions determine roots of...

CHAPTER 4Polynomial and Rational Functions

SECTION 4.1

Polynomial FunctionsDetermine roots of polynomial

equationsApply the Fundamental

Theorem of Algebra

POLYNOMIAL IN ONE VARIABLE

A polynomial in one variable x, is an expression of the form a0xn + a1xn-1 +….+ an-

1x + anx. The coefficients a0, a1,a2,…, an, represent complex numbers (real or imaginary), a0 is not zero and n represents a nonnegative integer. Example: 1000x18 + 500x10 + 250x5

Degree The greatest exponent of its variable

Leading Coefficient The coefficient with the greatest exponent

1000x18 + 500x10 + 250x5

Degree – 18, Leading Coefficient - 1000

POLYNOMIALS

Polynomial FunctionIf a function is defined by a

polynomial in one variable with real coefficients

F(x) =1000x18 + 500x10 + 250x5

ZerosThe values of x for a polynomial

function where f(x) = 0. Also known as the x-intercepts.

POLYNOMIALS Consider f(x) = x3 + -6x2 + 10x – 8

State the degree and leading coefficient.Degree of 3 and leading coefficient of 1

Determine whether 4 is a zero of f(x).Evaluate f(4)Yes it is a zero.

Example f(x) = 3x4 – x3 + x2 + x – 1State the degree and leading

coefficientDegree 4, leading coefficient of 3

Determine whether -2 is a zero of f(x)No it is not a zero of the polynomial

POLYNOMIALS Polynomial Equation

A polynomial that is set equal to zero Root

The solution for a polynomial equationZero and Root are often used

interchangeably but technically, you find the zero of a function and the root of an equation.

Can be an imaginary number Complex Numbers

Any number that can be written in the form a + bi, where a and b are real numbers and i is the imaginary unit

Pure Imaginary NumbersThe complex number a + bi when a = 0

and b does not equal 0 and i is the imaginary unit

POLYNOMIALS

Fundamental Theorem of AlgebraEvery polynomial equation with

degree greater than zero has at least one root in the set of complex numbers.

Corollary to the Fundamental Theorem of AlgebraStates that the degree of a

polynomial indicates the number of possible roots of a polynomial equation

POLYNOMIAL GRAPHS

Graphs on pg 207 Positive leading coefficients and degree

greater than 0 (Top Section) Shows maximum number of times the graph of

each type of polynomial may cross the x-axis General shape of a third degree function and

a fourth-degree function. (Bottom Section) The graph of a polynomial function with odd

degree MUST cross the x-axis AT LEAST ONCE The graph of a function with even degree

MAY or MAY NOT cross the x-axis; if it does it will an even number of times

POLYNOMIAL GRAPHS

Each x-intercept represents a real root of the polynomial equation

If a and b are roots of the equation, then using the corollary to the Fundamental Theorem of Algebra, we can find the polynomial equationSet equation up starting with (x-a)(x-

b)=0.

SECTION 4.2

Quadratic Equations Solve quadratic equations Use the discriminate to describe the roots of

quadratic equations

QUADRATIC EQUATIONS

A polynomial equation with a degree of two. Ways to Solve Quadratic Equations:

Graph Factor Completing the square

Completing the square Used to create a perfect square trinomial Useful when the equation can’t be factored (x + b)2= x2 + 2bx + b2

Given first and middle term, find last Square of half the coefficient of the middle term; only

works with the coefficient of the first term is 1

QUADRATIC EQUATIONS

Ex. x2 -6x -16 Graph

Look at x intercepts Factor

(x+2) and (x-8) Set equal to zero, x = -2, 8

Completing the square (x-3) 2 =25 x = -2, 8

Ex. 3x2 +7x + 7 Graph

Look at x intercepts No x intercepts; roots are imaginary numbers

Completing the square (x+ 7/6) 2 =-35/36 x = -7/6+/-i(35) 1/2 /6

QUADRATIC EQUATIONS

Quadratic Formula

Discriminant

-Tells the nature of the roots of a quadratic equation or the zeros of the related quadratic function

QUADRATIC EQUATIONS

b2-4ac >0 Two distinct real roots

b2-4ac=0 Exactly one real root (actually a double root)

b2-4ac<0 No real roots (Two distinct imaginary roots)

QUADRATIC EQUATIONS

Find the discriminant of x2 -4x +15 and describe the nature of the roots of the equation. Then find the roots. Discriminant = -44; D<0 no real roots Roots: 2-i(11) ½ and 2+i(11) ½

Conjugates Suppose a and b are real numbers with b not

equal to 0. If a + bi is a root of a polynomial equation with real coefficients, then a – bi is also a root of the equation. a + bi and a – bi are conjugate pairs

What are other examples? i and –i; -1 + i and -1 – i

QUADRATIC EQUATIONS

Solve 6x2 + x +2 by using graphing, factoring, completing the square, and the quadratic equation. Graphing

The graph does not touch the x-axis no real roots for the equation, can’t determine roots from graph

Factoring No real roots, factoring can’t be solved

Completing the square (x + 1/12) 2 = -47/144 Roots: -1+/-i(47) 1/2/12

Quadratic Equation A = 6, b = 1, c = 2 X= -1+/-i(47) 1/2/12

SECTION 4.3

The Remainder and Factor Theorems Find the factors of polynomials using the

Remainder and Factor Theorem

THE REMAINDER AND FACTOR THEOREMS

Quotient

Divisor Dividend

Remainder


Remainder Theorem If a polynomial P(x) is divided by x – r, the

remainder is a constant P(r), and P(x) =(x-r) * Q(x) + P(r),

where Q(x) is a polynomial with degree one less than the degree of P(x)


What is 2x2 + 3x -8 divided by x -2? Solve using long division Solve using synthetic 2x + 7 + 6/(x-2)

Divide x3 – x2 +2 by x +1? Solve using long division Solve using synthetic x2 -2x + 2


Factor Theorem The binomial x – r is a factor of the polynomial

P(x) if and only if P(r) = 0. IE. No remainder

Depressed Polynomial The quotient when a polynomial is divided by one of its

binomial factors x – r,

Ex: 2x3 – 3x2 +x divided by x-1 Is the quotient a factor and/or a depressed

polynomial? Yes it is both, 2x2 -x


Determine the binomial factors of x3 – 7x +6 using synthetic division

R 1 0 -7 6

-4

-3

-2

-1

0

1

2

R 1 0 -7 6

-4 1 -4 9 -30

-3 1 -3 -4 0

-2 1 -2 -3 12

-1 1 -1 -6 12

0 1 0 -7 6

1 1 1 -6 0

2 1 2 -3 0

Factors are:X+3, X-1, X-2


Determine the binomial factors of x3 – 7x +6 using the Factor Theorem Test values F(x) = x3 – 7x +6; Test -1

No because = 12 F(x) = x3 – 7x +6; Test 1

Yes works because = 0, then find depressed polynomial

Depressed polynomial is x2 + x -6 Now Factor depressed polynomial to get other factors Factors to (X-1)&(X-2) All Factors are (X+3),(X-1)&(X-2)


Determine the binomial factors of x3 -2x2-13x-10 X+1, X+2, X-5

Find the value of K so that the remainder of (x3 + 3x2 – kx – 24) divided by (x + 3) is

0. Set dividend equal to 0, plug in -3 for X, and then

solve for K K = 8 Check using synthetic division

SECTION 4.4THE RATIONAL ROOT THEOREM

Learn how to identify all possible rational roots of a polynomial equation using the rational root theorem

Determine the number of positive and negative real roots each polynomial function has

THE RATIONAL ROOT THEOREM

Let a0xn + a1xn-1 + …+ an-1x + an =0 represent a polynomial equation of degree n with integral coefficients. If a rational number p/q, where p and q have no common factors, is a root of the equation, then p is a factor of an and q is a factor of a0.

P is a factor of the last coefficient and Q is a factor of the first coefficient


List the possible roots of 6x3+11x2-3x-2=0 P must be a factor of 2 Q must be a factor of 6 Possible Values of P:

+/-1, +/-2 Possible Values of Q:

+/-1, +/-2, +/-3, +/-6 Possible rational roots, p/q :

+/-1, +/-2, +/-1/2, +/-1/3, +/-1/6, +/-2/3 Use graphing to narrow down the possibilities

Find zero at X = -2 Check using synthetic, then factor the depressed

polynomial to get roots X = -2, -1/3, 1/2


Integral Root Theorem Let xn + a1xn-1 + …+ an-1x + an =0 represent a

polynomial equation that has a leading coefficient of 1, integeral coefficients, and an can’t equal 0. Any rational roots of this equation must be integral factors of an.

Roots have to be a factor of an, the last coefficient

THE RATIONAL ROOT THEOREM Find the roots of x3+8x2+16x+5=0

How many roots are there? 3

What do they have to be factors of according to the integral root theorem? 5 Possible roots: +/-5 and +/-1

Do synthetic division with these roots to check which is a factor. (IE no remainder) Try 5

Doesn’t work, remainder of 410 Try -5

Works, no remainder Factor or use quadratic formula to find the roots of

the depressed polynomial. Roots: -5, -3-(5) 1/2/2, -3+(5)1/2/2


Descartes’ Rule of Signs Suppose P(x) is a polynomial whose terms are

arranged in descending powers of the variable. Then the number of POSITIVE real zeros

of P(x) is the same as the number of changes in sign of the coefficients of the terms or is less than this by an even number.

The number of NEGATIVE real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms in P(-x) or less than this number by an even number

THE RATIONAL ROOT THEOREM Find the number of possible positive real zeros and the

number of possible negative real zeros for f(x) = 2x5+3x4-6x3+6x2-8x+3 Positive Real Zeros:

4 Changes, 4, 2, or 0 possible positive real zeros Negative Real Zeros:

F(-x) = -2x5+3x4+6x3+6x2+8x+3

One change, 1 possible negative real zero Find Possible zeros

Possible Values of P: +/-1, +/-3

Possible Values of Q: +/-1, +/-2

Possible Values of P/Q: +/-1, +/-3, +/-1/2, +/-3/2

Test using synthetic division or graphing Rational Roots = -3, ½, 1

SECTION 4.5LOCATING ZEROS OF A POLYNOMIAL FUNCTION

Learn to approximate the real zeros of a polynomial function

LOCATING ZEROS OF A POLYNOMIAL FUNCTION

Location Principle Suppose y = f(x) represents a polynomial function with

real coefficients. If a and b are two numbers with f(a) negative and f(b) positive, the functions has at least one zero between a and b.

IE the answer of the equation at that root or the remainder changes signs between two roots.


Determine between which consecutive integers the real zeros of F(x) = x3 – 4x2 – 2x + 8 are located. Method 1: Synthetic Division

Test (-3, 5) Method 2: Graphing Calculator

Use Table Function There is a zero at 4, and between -2 and -1, and

between 1 and 2.


Approximate the real zeros of f(x) = 12x3-19x2-x+6 to the nearest tenth. How many zeros?

3 How many positive?

2 or 0 How many negative?

-12x3-19x2+x+6 1

Use graphing calculator Table to see where zeros fall Between -1 and 0, between 0 and 1, and between 1

and 2. Use graphing calculator TableSet to change delta from 1

to .1 to better see where 0’s fall Use graph to trace to see 0’s Zeros are at about -.5, .7, 1.4


Upper Bound Theorem Suppose c is a positive real number and P(x) is divided

by x – c. If the resulting quotient and remainder have no change in sign, then P(x) has no real zero greater than c. Thus c is an upper bound of the zeros of P(x).

Helps to determine if you have found all real zeros An integer greater than or equal to the greatest real

zero Lower Bound Theorem

If c is an upper bound of the zeros of P(-x), then –c is a lower bound of the zeros of P(x)

An integer less than or equal to the least real zero.


Find the upper and lower bound of the zeros of f(x) = x3 + 3x2-5x-10 Find real zeros:

-3.6, -1.4, 2 Interval of upper and lower bound?

-4<=x<=2 Find the upper and lower bound interval for f(x)

= 6x3-7x2-14x+15 -2 <=x<=3

SECTION 4.6RATIONAL EQUATIONS AND PARTIAL FRACTIONS

Learn how to solve rational equations and inequalities.

Learn how to decompose a fraction into partial fractions

SECTION 4.6RATIONAL EQUATIONS AND PARTIAL FRACTIONS

Rational Equation An equation with one or more rational expressions

What is a rational expression? The quotient of two polynomials in the form

f(x)=g(x)/h(x), where h(x) does not equal 0 How do you solve rational equations?

Multiply each side by the

RATIONAL EQUATIONS AND PARTIAL FRACTIONS

Example 1: Solve a2-5 = a2+a+2

a2-1 a+1 What is the LCD?

a2-1 What do we get for a?

a = 3 or -1 Can both of these be our answers?

a can only be 3 because if we plug in -1 to our original equation we get a denominator of 0.

Example 2: Solve X – 2 = 20 . X + 4 x – 1 x2 + 3x - 4 What is the LCD?

x2 + 3x – 4 which factors to (x-1) * (x+4) What is the answer?

7


Example 1: Decompose 8y + 7 into partial fractions. y2 + y - 2

First Factor Denominator (y-1) *(y+2)

Then split into two fractions on other side of equals 8y + 7 = A + B y2 + y – 2 (y-1) (y+2)

Multiply each piece by LCD to get rid of fractions 8y + 7 = A(y+2) + B(y-1)

Eliminate B by plugging in 1 for y Solve for A A = 3

Eliminate A by plugging in 2 for y Solve for B B = 3

Re-write fractions by plugging in values found for A and B 8y + 7 = 5 + 3 y2 + y – 2 (y-1) (y+2)

Check to see if the sum of the two fractions equal the original.


Example 2: Decompose 6x - 2 into partial fractions x2 -3x – 10

2 + 4 x+2 x-5


Rational Inequalities Same as equations but with inequality sign

Example 1: (x-2)(x-1) < 0 (x-3)(x-4)2

For what values is our domain undefined? 3 and 4

What values make this 0? 2 and 1

Plot these points on a number line with dashes at the above values What happens to our number line?

Splits into intervals Test each interval to see if our inequality is true or

false Works for intervals x <1 and 2<x<3

Show Solution on number line


Example 2: 2 + 5 > 3 3a 6a 4

Solve for a first, by multiplying by LCD of 12a. A = 2

What is the zero? 2

What is the excluded value? 0

Test intervals Works for intervals 0<a<2

Show Solution on number line

SECTION 4.7RADICAL EQUATIONS AND INEQUALITIES

Learn how to solve equations and inequalities with radicals involved.

RADICAL EQUATIONS AND INEQUALITIES

Radical Equations Equations in which radical expressions include

variables Extraneous Solutions

Solutions that do not satisfy the original equation Check all solutions back into original equation in

order to exclude those that don’t work

RADICAL EQUATIONS AND INEQUALITIES

Example: x = √x+7) +5 Solve for X x = 9 and x = 2 Check that neither are extraneous solutions Only 9 works, Answer: x=9

Example 2: 4 = 3√ x+2)+8 Solve for X x = -66 Check Works, Answer: x=-66

Example: √ x+1) = 1 + √ 2x-12) X = 8

RADICAL EQUATIONS AND INEQUALITIES Radical Inequalities

Same as equations but with inequality signs Example: √ 4x+5) <10

Solve for X X<23.75 Must also find the lower bound to make √ 4x+5) a real

number. Set √ 4x+5) =0 and solve X>-1.25 Solution is -1.25< X<23.75 Check by testing intervals Graph intervals on number line

Example: √ 6x-5) > 4 X > 7/2

c hapter 4 polynomial and rational functions. s ection 4.1 polynomial functions determine roots of...

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