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**Zeros of Polynomial Functions**

Take out your homework Warm-up Solve 3x3 – 2x2 + 3x – 4 ÷ x – 3 by Synthetic Division Find f(25) using synthetic substitution in the above expression List all the potential real factors of the expression in #1. Quiz Review:

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Warm-up Answers 45,696 ±1, ±2, ±4, ±4/3,±2/3, ±⅓

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**Learning Goals Fundamental Theorem of Algebra Conjugate Root Theorem**

Complex Zeros …be able to find a polynomial given zeros, real and complex …be able to factor and find complex zeros of a polynomial function

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**Fundamental Theorem of Algebra**

An nth-degree polynomial (n >0), has at least one zero in the complex number system. Corollary – An nth-degree polynomial function has exactly n zeros, including repeated zeros, in the complex number system.

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**Conjugate Root Theorem**

If a + bi (b ≠ 0), is a root of a polynomial function with real coefficients, then the complex conjugate, a - bi is also a root of the polynomial.

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**Find a Polynomial Function Given Its Zeros**

Write a polynomial function of the least degree possible, with real coefficients in standard form with the given zeros. Pg. 127: #35 (homework problem) **hint – Conjugate Root Theorem Solution: Zeros: -1, 8, (6 – i)

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Practice (10 min.) Pg. 127: #32, 33, 38, 85-86 Write the polynomial function in standard form, given the zeros. 3, -4, 6, -1 -2, -4, -3, 5 Use Synthetic Division

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**Wrap-up and Quiz Review**

I am able to find a polynomial given zeros, real and complex. Quiz Friday covering Long Division and Synthetic Division/Substitution.

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Quiz Review Take out your Quiz Review Worksheet.

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**Factoring Polynomials**

Every polynomial can be written as the product of linear factors and/or irreducible quadratic factors, each with real coefficients. A quadratic factor is irreducible over the reals when it has real coefficients, but no real zeros associated with it. Example: (x2+4)

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**Factor and Find the Zeros of a Polynomial Function**

Starting point? Total number of Zeros Rational Zero Theorem – List all possible Rational Zeros Graph to find likely real zeros Use Synthetic Division to: Check for factors factor the function

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**Example Factor and find the zeros of the equation: Total zeros – 5**

RZT – All possible rational zeros – {±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30} Find potential zeros on a graph that are in the list.

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**Choose possible zeros to try with synthetic substitution.**

Example continued: Choose possible zeros to try with synthetic substitution. (-5, 2, 3 are all good possibilities) -5 1 -18 30 -19 25 -35 -30 7 6 2 1 -5 7 6 -6 -3 3 1 -3 𝑥 2 +1

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**Factor polynomial – all linear factors**

Continue from the previous Example: 𝑓 𝑥 = 𝑥+5 𝑥−2 𝑥−3 𝑥 2 +1 Rewrite all irreducible factors as complex linear factors. Final answer written in all linear factors

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Wrap-up I am able to factor and find complex zeros of a polynomial function.

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**Practice Pg. 127: #42-44, Complex Zeros WS**

42-44 – write as a) the product of irreducible linear factors, b) the product of linear factors, and c) list all the zeros (Real and Complex)

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