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Finding Real Roots of Polynomial Equations • How do we identify the multiplicity of roots? • How do we use the Rational Root Theorem and the irrational Root Theorem to solve polynomial equations? Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations In Lesson 3-5, you used several methods for factoring polynomials. As with some quadratic equations, factoring a polynomial equation is one way to find its real roots. Using the Zero Product Property, you can find the roots, or solutions, of the polynomial equation P(x) = 0 by setting each factor equal to 0 and solving for x. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Example 1: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. 4x6 + 4x5 – 24x4 = 0 Check using a graph. 2 4 4x x x 6 0 4 x x 3 x 2 0 4 4x 0 x 3 0 x 2 0 4 x0 x 3 x2 The roots appear to be located at x = 0, x = –3, and x = 2. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Example 2: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. x4 + 25 = 26x2 x 26 x 25 0 4 2 x 25 x 1 0 x 5 x 5 x 1 x 1 0 2 2 x 5 0 x 5 0 x 1 0 x 1 0 x 5 x5 x 1 The roots are 5, –5, 1, and –1. Holt McDougal Algebra 2 x 1 Finding Real Roots of Polynomial Equations Example 3: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. 2x6 – 10x5 – 12x4 = 0 2 x x 5 x 6 0 4 2 x x 6 x 1 0 4 2 2x 0 x 6 0 x 1 0 4 x0 x6 x 1 The roots are 0, 6, and –1. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Example 4: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. x3 – 2x2 – 25x = –50 3 2 x 2 x 25 x 50 0 x x 2 25 x 2 0 2 x 2 25 x 2 0 x 5 x 5 x 2 0 x5 0 x5 0 x2 0 x5 x 5 Holt McDougal Algebra 2 x2 The roots are 5, –5, and 2. Finding Real Roots of Polynomial Equations Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root. In 3x5 + 18x4 + 27x3 = 0 has two multiple roots, 0 and –3. For example, the root 0 is a factor three times because 3x3 = 0. The multiplicity of root r is the number of times that x – r is a factor of P(x). When a real root has even multiplicity, the graph of y = P(x) touches the x-axis but does not cross it. When a real root has odd multiplicity greater than 1, the graph “bends” as it crosses the x-axis. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations You cannot always determine the multiplicity of a root from a graph. It is easiest to determine multiplicity when the polynomial is in factored form. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Example 5: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. 2x4 12x3 + 18x2 = 0 2x x 6 x 9 0 2 2 x x 3 x 3 0 2 2x 0 x 3 0 x 3 0 2 x0 2 x3 x3 The root 0 has multiplicity of 2, the root 3 has multiplicity of 2. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Example 6: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. x3 – x2 – x + 1 = 0 2 x x 1 1 x 1 0 x 1 x 1 0 2 x 1 x 1 x 1 0 x 1 0 x 1 0 x 1 0 x 1 x 1 x 1 The root 1 has multiplicity 2, the root 1 has multiplicity 1. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Lesson 4.1 Practice A Holt McDougal Algebra 2