7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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Presentation transcript:

7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write a polynomial function. Standard: N. Solve equations both symbolically and graphically.

The Rational Root Theorem can be used to identify possible roots of polynomial equations with integer coefficients. Rational Root Theorem Let P be a polynomial function with integer coefficients in standard form. If p/q (in lowest terms) is a root of P(x) = 0, then p is a factor of the constant term of P and q is a factor of the leading coefficient of P.

* 8x x x + 2 = 0

* Q(x) = x 3 - 6x 2 + 7x + 2

* Q(x) = x 3 + 4x 2 – 6x – 12

Ex 3. Find all of the zeros of: Same as above, but you will get an imaginary #. * P(x) = 3x 3 – 10x x – 4 * P(x) = 3x 3 – 10x x – 4

* P(x) = x 3 - 9x x – 145

* P(x) = -4x 3 + 2x 2 – 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 Activities

PSSA Warm-Up Question Algebra II Chp 7 Standard S Analyze linear, polynomial, and rational functions. How can you identify and describe functions and their graphs? What are the functions zero(s)? 1). Linear Function y = ½x + 2 2). Quadratic Function y = x 2 – 2x – 3 3). Cubic Function y = x 3 – 4x

Review of Zeros of Polynomial Functions

Homework Integrated Algebra II- Section 7.5 Level A Academic Algebra II- Section 7.5 Level B