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1 Algebra II – Chapter 6 Day #5 Topic: Theorems About Roots of Polynomial Equations Today, we will focus our attention on theorems that will help us solve polynomial equations. Standards/Goals: N.CN.7.: I can solve quadratic equations with real coefficients that have complex solutions. I can use the Rational Root Theorem to solve equations. I can use the Conjugate Root Theorem to solve equations. I can use the Descartes’ Rule of Signs to determine the number of roots of a polynomial equation. I can use synthetic division to divide two polynomials. We want to first look at the Rational Root Theorem Rational Root Theorem 𝑝 This theorem says that any rational zero of a function must be in the form of 𝑞, where ‘p’ is a factor of the constant term and ‘q’ is a factor of the leading coefficient. This theorem will NOT help us find irrational or imaginary zeros; to find these we must use the reduced polynomial and quadratic formula. EXAMPLES: List all of the potential rational zeros of the following polynomials. Then use polynomial division and the quadratic formula to identify the actual zeros. 1. f(x) = 3𝑥 3 + 5𝑥 2 − 26𝑥 + 8 2. p(x) = 𝑥 4 − 5𝑥 3 + 10𝑥 2 − 20𝑥 + 24 We also want to familiarize ourselves with the CONJUGATE ROOTS THEOREM. Conjugate Roots Theorem This theorem basically says that when we consider a function that is a polynomial with only real coefficients, if a + bi is a zero of ‘f’, then so is the complex number a – bi. In terms of the linear factors of ‘f’, this means that x – (a + bi) is a factor of f, then so is x – (a – bi). EXAMPLES: #1. A quartic polynomial P(x) has rational coefficients. If 2 – i and −√5 are roots of P(x) = 0, what are the other roots? #2. A cubic polynomial P(x) has rational coefficients. If 2 + 3i and 2/3 are two roots of P(x) = 0, what is one additional root? 2 #3. What is a quartic polynomial function P(x) with rational coefficients so that P(x) = 0 has roots 1, 3, and 1 + 2i? #4. A cubic polynomial P(x) with rational coefficients so that P(x) = 0 has roots -3 and i? EXAMPLES: Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to factor the following polynomial equations completely, making use of the given zero). #1. f(x) = 𝑥 4 + 3𝑥 3 + 21𝑥 2 + 147𝑥 − 1372; -7i is a zero #2. p(x) = 𝑥 4 − 2𝑥 3 + 14𝑥 2 − 8𝑥 + 40; 2i is a zero. We will also use the Descartes’ Rule of Signs to determine how many roots there are in a polynomial. Descartes’ Rule of Signs Let P(x) be a polynomial with real coefficients written in standard form. The number of positive real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(x) or is less than that by an even number. The number of negative real roots of P(x) = 0 is either equal to the number of sign changes between consecutive coefficients of P(-x) or is less than that by an even number. In both cases, count multiple roots according to their multiplicity. EXAMPLES: #1. What does Descartes’ Rule of Signs tell you about the real roots of 𝑥 3 − 𝑥 2 + 𝑥 + 1 = 0? #2. What does Descartes Rule of Signs tell you about the real roots of −𝑥 4 + 𝑥 3 − 2𝑥 2 + 𝑥 + 1 = 0? 3 HOMEWORK – Chapter 6 Day #5 Use the Rational Root Theorem to list all possible rational roots for each equation. 1. x3 + 5x2 2x 15 = 0 2. 36x3 + 144x2 x 4 = 0 Use the Rational Root Theorem to list all possible rational roots for each equation. Then find any actual rational roots. 3. 2x3 + 5x2 + 4x + 1 = 0 4. 12x4 + 14x3 5x2 14x 4 = 0 A polynomial function P(x) with rational coefficients has the given roots. Find two additional roots of P(x) = 0. 5. 2 + 3i and 7 6. 3 2 and 1 3 7. 4i and 6 i Write a polynomial function with rational coefficients so that P(x) = 0 has the given roots. 8. 4 and 6 9. 5 and 1 10. 3i and 6 11. 2 + i and 1 5 12. 5 and 3i EXAMPLES: Use all available methods (in particular, the Conjugate Roots Theorem, if applicable) to factor the following polynomial equations completely, making use of the given zero). 13. 𝑓(𝑥) = 𝑥 4 − 5𝑥 3 + 19𝑥 2 − 125𝑥 − 150; -5i is a zero. 14. 𝑓(𝑥) = 𝑥 4 + 7𝑥 3 + 46𝑥 2 + 252𝑥 + 360; −6𝑖 𝑖𝑠 𝑎 𝑧𝑒𝑟𝑜. What does Descartes’ Rule of Signs say about the number of positive real roots and negative real roots for each polynomial function? 15. P(x) = 3x3 + x2 8x 12 16. P(x) = 2x4 x3 3x + 7 17. P(x) = 4x5 x4 x3 + 6x2 5