6.2 Polynomials and Linear Factors

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Presentation transcript:

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

Factoring the Sums and Differences of Cubes 13 = _ 23 = _ 33 = _ 43 = _ 53 = _ 63 = _ 73 = _ 83 = _ 93 = _ 103 = _ 1 8 27 64 125 216 343 512 729 1000 Being familiar with perfect cubes will make cubing binomials and factoring sums and differences of cubes easier.

Special Factoring Patterns …more know-by-♥ stuff! Special Factoring Patterns Sum of Two Cubes u3 + v3 = Example: u3 + 27 = Difference of Two Cubes u3 - v3 = u3 + 64 = (u + v) (u2 - uv + v2) (u - v) (u2 + uv + v2)

Factor the polynomials: a. x3 - 27 b. x3 - 64 c. x3 + 125 d. x3 + 216 e. 4x3 + 500 f. 3x 3 - 192

REVIEW: FACTORING BY GROUPING Factor the polynomials: a. 4x3 - 6x2 + 10x – 15 b. x3 - 2x2 - 4x + 8

REVIEW: Factoring By Using Combined Methods Factor the polynomial COMPLETELY: 2x3 + 10x2 + 12X

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!

SOLVING POLYNOMIAL EQUATIONS BY FACTORING Solve the following equations by factoring: a. 3x4 = 48x2 b. x3 - 216 c. 5x3 + 135

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

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.

Final Checks for Understanding Given x3 - 4x2 = - 4x, determine the following: Factors: __________________ Solutions: X-intercepts: Zeros:

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