Chapter 6: Polynomials and Polynomial Functions Section 6.2: Polynomials and Linear Factors Content Objectives: Students will demonstrate analysis of polynomial functions by simplifying them into their factored form to obtain the zeros. Language Objectives: Students will write a paragraph explaining how the graph of a polynomial function can help you factor that polynomial.

Linear Factor: similar to a prime number in that it can not be factored any further. Example 1: Write as a polynomial in standard form. a)(x - 8)(x + 3)(x + 9) Page 313 Quick check #1 (x + 1)(x + 1)(x + 2)

Example 2: Write in completed factored form, (Always check for GCF) a)3x^3 - 18x^2 + 24x b) y= 2x^3 - 18x^2 + 16x Page 314 Quick Check #2 6x^3 - 5x^2 - 36x

Recall: If a polynomial is in factored form, you can use the Zero Product Property to find values that will make the polynomial equal zero.

Example 4: find the zeros and sketch the function a)y = (x - 8)(x + 3)(x + 9) b) y =(x - 2)(x + 9) c) y = (x - 7)(x + 1)(x - 4).

Theorem: Factor Theorem The expression x a is a linear factor of a polynomial if and only if the value of "a" is a zero of the related polynomial function. Example 5: Write a polynomial function in standard form with: a)zeros at 2, -3, and 0 b) zeros at 4, -2, and 1

If a linear factor of a polynomial is repeated, then the zero is repeated. A repeated zero is called a multiple zero. A multiple zero has multiplicity equal to the number of times the zero occurs. Example 6: Find any multiple zeros and state the multiplicity. a)f(x)= x^5 6x^4 + 9x^3 b) f(x) = (x - 2)(x + 1)(x + 1)^2

Textbook Assignment: Pg #’s 1-13, 17-35, [odds]