7.5 zeros of polynomial functions objectives: use the rational root theorem and the complex...
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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