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Theorems About Roots of Polynomial Equations

1. Use the Rational Root Theorem to list all possible rational roots of 3 x 3 + x 2 – 15 x – 5 = 0. Then find any actual rational roots. 2. Find the roots of 10 x 4 + x 3 + 7 x 2 + x – 3 = 0. 1 3. 5 3. 1 3. ±1, ±5, ± , ± ; –. 3 5. 1 2. – , , ± i.

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Theorems About Roots of Polynomial Equations

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  1. 1. Use the Rational Root Theorem to list all possible rational roots of 3x3 + x2 – 15x – 5 = 0. Then find any actual rational roots. 2. Find the roots of 10x4 + x3 + 7x2 + x – 3 = 0. 1 3 5 3 1 3 ±1, ±5, ± , ± ; – 3 5 1 2 – , , ±i Theorems About Roots of Polynomial Equations ALGEBRA 2 LESSON 6-5 6-5

  2. A polynomial equation with rational coefficients has the roots 2 – 5 and 7 . Find two additional roots. By the Irrational Root Theorem, if 2 – 5 is a root, then its conjugate 2 + 5 is also a root. If 7 is a root, then its conjugate – 7 also is a root. Theorems About Roots of Polynomial Equations ALGEBRA 2 LESSON 6-5 6-5

  3. Theorems About Roots of Polynomial Equations ALGEBRA 2 LESSON 6-5 A polynomial equation and real coefficients has the roots 2 + 9i with 7i. Find two additional roots. By the Imaginary Root Theorem, if 2 + 9i is a root, then its complex conjugate 2 – 9i also is a root. If 7i is a root, then its complex conjugate –7i also is a root. 6-5

  4. Theorems About Roots of Polynomial Equations ALGEBRA 2 LESSON 6-5 Find a third degree polynomial with rational coefficients that has roots –2, and 2 – i. Step 1: Find the other root using the Imaginary Root Theorem. Since 2 – i is a root, then its complex conjugate 2 + i is a root. Step 2: Write the factored form of the polynomial using the Factor Theorem. (x + 2)(x – (2 – i))(x – (2 + i)) 6-5

  5. Theorems About Roots of Polynomial Equations ALGEBRA 2 LESSON 6-5 (continued) Step 3: Multiply the factors. (x + 2)[x2 – x(2 – i) – x(2 + i) + (2 – i)(2 + i)] Multiply (x – (2 – i)) (x – (2 + i)). (x + 2)(x2 – 2x + ix – 2x – ix + 4 – i2) Simplify. (x + 2)(x2 – 2x – 2x + 4 + 1) (x + 2)(x2 – 4x + 5) Multiply. x3 – 2x2 – 3x + 10 A third-degree polynomial equation with rational coefficients and roots –2 and 2 – i is x3 – 2x2 – 3x + 10 = 0. 6-5

  6. Assignment 47 • Pg 333 14-24,29-36,40,42,43

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