Essential Questions How do we identify the multiplicity of roots?

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Presentation transcript:

Essential Questions 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?

In Lesson 3-5, you used several methods for factoring polynomials 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.

Example 1: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. 4x6 + 4x5 – 24x4 = 0 Check using a graph. The roots appear to be located at x = 0, x = –3, and x = 2. 

Example 2: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. x4 + 25 = 26x2 The roots are 5, –5, 1, and –1.

Example 3: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. 2x6 – 10x5 – 12x4 = 0 The roots are 0, 6, and –1.

Example 4: Using Factoring to Solve Polynomial Equations Solve the polynomial equation by factoring. x3 – 2x2 – 25x = –50 The roots are 5, –5, and 2.

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.

You cannot always determine the multiplicity of a root from a graph 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.

Example 5: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. 2x4 - 12x3 + 18x2 = 0 The root 0 has multiplicity of 2, the root 3 has multiplicity of 2.

Example 6: Identifying Multiplicity Identify the roots of each equation. State the multiplicity of each root. x3 – x2 – x + 1 = 0 The root 1 has multiplicity 2, the root -1 has multiplicity 1.

Lesson 4.1 Practice A