11

Polynomial FunctionsPolynomial Functions

Exploring Polynomial FunctionsExploring Polynomial Functions– ExamplesExamples

Modeling Data with Polynomial Modeling Data with Polynomial FunctionsFunctions– ExamplesExamples

22

DegreeDegree Name of Name of DegreeDegree

Number Number of Termsof Terms

Name Name using using

number of number of termsterms

00 ConstantConstant 11 MonomialMonomial

11 LinearLinear 22 BinomialBinomial

22 QuadraticQuadratic 33 TrinomialTrinomial

33 CubicCubic 44 33rdrd degree degree

PolynomialPolynomial

44 QuarticQuartic nn 44thth degree degree Polynomial Polynomial

with n termswith n terms

55 QuinticQuintic 55thth degree degree polynomial polynomial

with n termswith n terms

33

xx 00 33 55 66 99 1111 1212 1414

yy 4242 3131 2626 2121 1717 1515 1919 2222

44

55

Polynomials and Linear Polynomials and Linear FactorsFactors Standard FormStandard Form

– ExampleExample Factored FormFactored Form

– ExamplesExamples Factors and ZerosFactors and Zeros

– ExamplesExamples

66

Writing a polynomial in standard form

You must multiply:

(x + 1)(x+2)(x+3)

X3 + 6x2 + 11x + 6

77

2x3 + 10x2 + 12x

2x(x2 + 5x +6)

88

The expression x-a is a linear factor of a polynomial if and only if thevalue a is a zero of the related polynomial function.

Factor TheoremFactor Theorem

99

Factors and ZerosFactors and Zeros

-3-3 -2-2 -1-1 00 11 22 33

(x – (-3)) or (x + 3)(x – (-3)) or (x + 3) (x – (-2)) or (x + 2)(x – (-2)) or (x + 2) (x – (-1)) or (x + 1)(x – (-1)) or (x + 1) (x – 0) or x(x – 0) or x (x – 1)(x – 1) (x – 2)(x – 2) (x – 3)(x – 3)

ZEROS FACTORS

1010

Dividing PolynomialsDividing Polynomials

Long DivisionLong Division Synthetic DivisionSynthetic Division

1111

Long DivisionLong Division

The purpose of this type of division The purpose of this type of division is to use one factor to find is to use one factor to find another.another.

)4 40

Just as 4 finds the 10

)x - 1 x3 + 6x2 -6x - 1

The (x-1) finds the (x2 + 7x + 1)

1212

Synthetic DivisionSynthetic Division

When dividing by x – a, use When dividing by x – a, use synthetic division.synthetic division.

The Remainder TheoremThe Remainder Theorem

1313

The Remainder The Remainder TheoremTheorem When using Synthetic Division, When using Synthetic Division,

the remainder is the value of f(a).the remainder is the value of f(a). This method is as good as This method is as good as

“PLUGGING IN”, but may be “PLUGGING IN”, but may be faster.faster.

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Solving Polynomial Solving Polynomial EquationsEquations Solving by GraphingSolving by Graphing Solving by FactoringSolving by Factoring

1515

Solving by GraphingSolving by Graphing

Set equation equal to 0, then Set equation equal to 0, then substitute y for 0. Look at the x-substitute y for 0. Look at the x-intercepts. (Zeros)intercepts. (Zeros)

Let the left side be yLet the left side be y11and let the and let the right side be yright side be y22. (Very much like . (Very much like solving a system of equations by solving a system of equations by graphing). Look at the points of graphing). Look at the points of intersection.intersection.

1616

Solving by FactoringSolving by Factoring

Sum of two cubesSum of two cubes

(a(a33 + b + b33) = (a + b)(a) = (a + b)(a22 – ab + b – ab + b22))

Difference of two cubesDifference of two cubes

(a(a33 – b – b33) = (a – b)(a) = (a – b)(a22 + ab + b + ab + b22))

1717

More on FactoringMore on Factoring

If a polynomial can be factored If a polynomial can be factored into linear or quadratic factors, into linear or quadratic factors, then it can be solved using then it can be solved using techniques learned from earlier techniques learned from earlier chapters.chapters.

Solving a polynomial of degrees Solving a polynomial of degrees higher than 2 can be achieved by higher than 2 can be achieved by factoring.factoring.

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Theorems about RootsTheorems about Roots

Rational Root TheoremRational Root Theorem Irrational Root TheoremIrrational Root Theorem Imaginary Root TheoremImaginary Root Theorem

1919

Rational Root TheoremRational Root Theorem

What are Rational Roots?What are Rational Roots? P’s and Q’s ………. ;)P’s and Q’s ………. ;) Using the calculator to speed up Using the calculator to speed up

the process.the process.

2020

……And the Rational And the Rational Roots are…..Roots are…..

q

p P includes all of the factors of the constant.

Q includes all of the factorsof the leading coefficient.

f(x) = x3 – 13x - 12

p = 12q = 1

The possible rational roots are:

1,2,3,4,6,12

2121

Test the Possible Test the Possible Roots…Roots…

In this case all roots are real and rational, but you need onlyto find one rational root. This will become clear later.

2222

Since -1, -3, and 4 are the Roots,

(x + 1), (x + 3), and (x – 4)

are the factors.

Multiply to show that

(x+1)(x+3)(x-4) = x3 – 13x – 12

(x+1)(x2 – x – 12)

x3 – x2 – 12x +x2 – x – 12

x3 – 13x – 12

2323

Irrational Root Irrational Root TheoremTheorem

If is a root, then is too.

ba ba

These are called CONJUGATES.

2424

Imaginary Root Imaginary Root TheoremTheorem

These are called CONJUGATES.

If is a root, then is too.

bia bia

2525

The Fundamental The Fundamental Theorem of AlgebraTheorem of Algebra If P(x) is a polynomial of degree If P(x) is a polynomial of degree

with complex coefficients, with complex coefficients, then P(x) = 0 has at least one then P(x) = 0 has at least one complex root. complex root.

A polynomial equation with degree A polynomial equation with degree n n will have exactly will have exactly nn roots; the roots; the related polynomial function will related polynomial function will have exactly have exactly nn zeros. zeros.

1n

2626

The Binomial TheoremThe Binomial Theorem

Binomial Expansion and Pascal’s Binomial Expansion and Pascal’s TriangleTriangle

The Binomial TheoremThe Binomial Theorem

2727

11 1

1 2 1 1 3 3 1

1 4 6 4 1 1 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1

PASCAL’S TRIANGLEPASCAL’S TRIANGLE