6.2 polynomials and linear factors at the end of this lesson, you should be able to: factor sums...

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6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation by factoring Write a polynomial function from its zeros

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Page 1: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

6.2 Polynomials and Linear Factors

At the end of this lesson, you should be able to:

Factor sums and differences of cubes Solve a polynomial equation by factoring Write a polynomial function from its zeros

Page 2: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

Factoring the Sums and Differences of Cubes

13 = _ 23 = _ 33 = _ 43 = _ 53 = _

63 = _ 73 = _ 83 = _ 93 = _ 103 = _

1 64278

216 512 1000

125

729343

Being familiar with perfect cubes will make cubing binomials and factoring sums and

differences of cubes easier.

Page 3: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

Special Factoring Patterns

Sum of Two Cubesu3 + v3 =

Example: u3 + 27 =

Difference of Two Cubesu3 - v3 =

Example: u3 + 64 =

(u + v) (u2 - uv + v2)

(u - v) (u2 + uv + v2)

…more know-by-♥ stuff!

Page 4: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

Factor the polynomials:

a. x3 - 27 b. x3 - 64

c. x3 + 125 d. x3 + 216

e. 4x3 + 500 f. 3x 3 - 192

Page 5: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

REVIEW: FACTORING BY GROUPING

Factor the polynomials:

a. 4x3 - 6x2 + 10x – 15

b. x3 - 2x2 - 4x + 8

Page 6: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

REVIEW: Factoring By Using Combined Methods

Factor the polynomial COMPLETELY:

2x3 + 10x2 + 12X

Page 7: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

ZERO PRODUCT PROPERTY

Recall: Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.

Alert!

This property connects factoring to solving equations!

Page 8: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

SOLVING POLYNOMIAL EQUATIONS BY FACTORING

Solve the following equations by factoring:

a. 3x4 = 48x2 b. x3 - 216 c. 5x3 + 135

Page 9: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

FACTOR THEOREM

The expression x – a is a linear factor of a polynomial if and only if the value a is a zero of the polynomial function.

You can reverse the process and write the linear factors when you know the zeros (solutions). The relationship between the linear factors of a polynomial and the zeros of a polynomial is given by the Factor Theorem.

Write the factors of a polynomial whose zeros are 2, 2, & -3.

Page 10: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

WRITING A POLYNOMIAL FUNCTION FROM ITS ZEROS

Alert!

A repeated zero is called a multiple zero. A multiple zero has a multiplicity equal to the number of times the zero occurs.

What is the multiplicity of the polynomial function? Explain.

Page 11: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

Final Checks for Understanding

Given x3 - 4x2 = - 4x, determine the following:

Factors:

__________________

Solutions:

__________________

X-intercepts:

__________________

Zeros: __________________

Page 12: 6.2 Polynomials and Linear Factors At the end of this lesson, you should be able to: Factor sums and differences of cubes Solve a polynomial equation

Homework

Connections: Zeros, Factors, and Solutions WS, plus text_______________

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