Section 3-5Finding Real Roots of Polynomial Equations

Objectives Identify the multiplicity of roots

Use the Rational Root Theorem and the Irrational Root Theorem to solve polynomial equations

As with some quadratic equations, factoring a polynomial equation is one way to find its real roots

Factoring Polynomials in Quadratic Form (Degree 4 or Higher)◦ Follow previous factoring rules◦ Remember rules of exponents when creating

binomials Recall 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

Finding Real Roots of Polynomial Equations

4x6 + 4x5 – 24x4 = 0

x3 – 2x2 – 25x = –50

Solve the Polynomial Equation by Factoring

Sometimes a polynomial equation has a factor that appears more than once. This creates a multiple root.

In the root, 0, is a multiple root. ◦ For example, the root of 0 is a factor FOUR times

because . 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

Finding Real Roots of Polynomial Equations

You can use a graphing calculator to see roots You cannot always determine the multiplicity

of a roots from a graph. It is easiest to determine multiplicity when

the polynomial is in factored form. If you can’t factor, use synthetic division to

help you. Get the root for synthetic from the graphing

calculator

Finding the Real Roots of Polynomial Equations

x3 + 6x2 + 12x + 8 = 0

x4 + 8x3 + 18x2 – 27 = 0

x4 – 8x3 + 24x2 – 32x + 16 = 0

Identify the roots of each equation. State the multiplicity of each root

Not all polynomials are factorable, but the Rational Root Theorem can help you find all possible rational roots of a polynomial equation.◦ Also known as the “p over q” theorem

Finding Real Roots of Polynomial Equations

Use the Rational Root Theorem to list all possible rational roots.

Rational Root Theorem

The design of a box specifies that its length is 4 inches greater than its width. The height is 1 inch less than the width. The volume of the box is 12 cubic inches. What is the width of the box?

Marketing Application

Polynomial equations may also have irrational roots

Finding real Roots of Polynomial Equations

The Irrational Root Theorem says that irrational roots come in conjugate pairs◦ For example, if you know that is a root of x3 – x2

– 3x – 1 = 0, then you know that is also a root Recall that the real numbers are made up of

the rational and irrational numbers. You can use the Rational Root Theorem and the Irrational Root Theorem together to find all of the real roots of P(x) = 0

Finding Real Roots of Polynomial Equations

2x3 – 9x2 + 2 = 0

2x3 – 3x2 –10x – 4 = 0

Identify all of the Real Roots of a Polynomial Equation

DAY 1: p186 #15-26 Day 2: p187 #29-35, 40-44

Page186-188 #15-22, 24-35, 40-43

Accelerated: ADD #44-47

HOMEWORK