7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex...

7.5 Zeros of 7.5 Zeros of Polynomial Polynomial

Functions Functions

Objectives:Objectives: Use the Rational Root Use the Rational Root Theorem and the Complex Conjugate Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Root Theorem. Use the Fundamental

Theorem to write a polynomial function.Theorem to write a polynomial function.

Standard:Standard: 2.8.11.N. Solve equations 2.8.11.N. Solve equations both symbolically and graphically.both symbolically and graphically.

The Rational Root Theorem can be used to identify possible roots of

polynomial equations with integer coefficients.

Rational Root TheoremLet P be a polynomial function with integer

coefficients in standard form. If p/q (in lowest terms) is a root of P(x) =

0, thenp is a factor of the constant term of P andq is a factor of the leading coefficient of P.

* 8x3 + 10x2 - 11x + 2 = 0

* * Q(Q(xx) = ) = xx33 - 6- 6xx22 + 7+ 7xx + + 22

* Q(x) = x3 + 4x2 – 6x – 12

Ex 3. Find all of the zeros of:Ex 3. Find all of the zeros of: Same Same as above, but you will get an as above, but you will get an imaginary #.imaginary #.

* * P(P(xx) = 3) = 3xx 3 3 – 10– 10xx 2 2 + 10+ 10xx – 4 – 4

* P(x) = x3 - 9x2 + 49x – 145

* P(x) = -4x 3 + 2x2 – x + 3

Complex Conjugate Root Theorem

If P is a polynomial function with real-number

coefficients and a + bi (where b ≠ 0) is a root of P(x) = 0, then a – bi is also a root of P(x) = 0.

Fundamental Theorem of Algebra: Every polynomial function of degree n ≥ 1 has at least one complex zero

Corollary: Every polynomial function of degree n ≥ 1 has exactly n complex zeros,

counting multiplicities.

Writing ActivitiesWriting Activities

PSSA Warm-Up QuestionPSSA Warm-Up QuestionAlgebra II Chp 7Algebra II Chp 7

Standard 2.8.11 S Analyze linear, polynomial, and rational functions.Standard 2.8.11 S Analyze linear, polynomial, and rational functions.

How can you identify and describe functions and their graphs? What How can you identify and describe functions and their graphs? What are the functions zero(s)?are the functions zero(s)?

1). Linear Function y = ½x + 21). Linear Function y = ½x + 2

2). Quadratic Function y = x2). Quadratic Function y = x22 – 2x – 3 – 2x – 3

3). Cubic Function y = x3). Cubic Function y = x33 – 4x – 4x

Review of Zeros of Review of Zeros of Polynomial FunctionsPolynomial Functions

Homework

Integrated Algebra II- Section 7.5 Level A

Academic Algebra II- Section 7.5 Level B

7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex...

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