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

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

Page 1: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 2: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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.

Page 3: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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.

Page 4: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 5: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 6: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 7: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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

Page 8: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 9: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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

Page 10: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

* Q(x) = x3 + 4x2 – 6x – 12

Page 11: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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

Page 12: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

* P(x) = x3 - 9x2 + 49x – 145

Page 13: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

* P(x) = -4x 3 + 2x2 – x + 3

Page 14: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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.

Page 15: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 16: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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.

Page 17: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 18: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

Writing ActivitiesWriting Activities

Page 19: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

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

Page 20: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

Review of Zeros of Review of Zeros of Polynomial FunctionsPolynomial Functions

Page 21: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.
Page 22: 7.5 Zeros of Polynomial Functions Objectives: Use the Rational Root Theorem and the Complex Conjugate Root Theorem. Use the Fundamental Theorem to write.

Homework

Integrated Algebra II- Section 7.5 Level A

Academic Algebra II- Section 7.5 Level B


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math1005postexam2reviewsolnF13 (1) Page 1quick049/Postexam2ReviewKEY.pdf5. Use the rational zeros theorem to determine the potential rational zeros Of the polynomial function f(x)

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