Chapter 3

CHAPTER 3 3-5 Finding the real roots of polynomial Equations

SAT Problem of the day • If f(3)=9 ,then

• A)-2• B)0• C)2• D)16• E)can not be determined

Solution to the SAT Problem • Right Answer: E

objectivesIdentify the multiplicity of roots.Use the Rational Root Theorem and the irrational Root Theorem to solve polynomial equations.

Real Zeros • In previous lessons, you used several methods for

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

• 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.

Example#1• Solve the polynomial equation by factoring• 4x6 + 4x5 – 24x4 = 0

Example#2• Solve the polynomial equation by factoring. • x4 + 25 = 26x2

Example#3• Solve the polynomial equation by factoring. • 2x6 – 10x5 – 12x4 = 0

Student guided practice• Do problems 2-5 in your book page 186

Real Zeros• 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.

Multiplicity • What is multiplicity?• Answer: 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.

Example

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#4• Identify the roots of each equation. State the

multiplicity of each root.• x3 + 6x2 + 12x + 8 = 0• Solution:• x3 + 6x2 + 12x + 8 = (x + 2)(x + 2)(x + 2)• x + 2 is a factor three times. The root –2 has a multiplicity

of 3.

Example#5• Identify the roots of each equation. State the

multiplicity of each root.• x4 + 8x3 + 18x2 – 27 = 0• Solution:• x4 + 8x3 + 18x2 – 27 = (x – 1)(x + 3)(x + 3)(x + 3)• x – 1 is a factor once, and x + 3 is a factor three times.

The root 1 has a multiplicity of 1. The root –3 has a multiplicity of 3.

Student guided practice• Do problems 8 and 9 in your book page 186

Rational Theorem• Not all polynomials are factorable, but the Rational Root

Theorem can help you find all possible rational roots of a polynomial equation.

• What is the rational zero?

Application• 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?

Application• A shipping crate must hold 12 cubic feet. The width

should be 1 foot less than the length, and the height should be 4 feet greater than the length. What should the length of the crate be?

Irrational root theorem• Polynomial equations may also have irrational roots.

Irrational root theorem• The Irrational Root Theorem say that irrational roots come

in conjugate pairs. For example, if you know that 1 + is a root of x3 – x2 – 3x – 1 = 0, then you know that 1 – is also a root.

22

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.

Example• Identify all the real roots of 2x3 – 9x2 + 2 = 0.

Example• Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0.

Student guided practice• Do problems 11-14 in your book page 186

Homework!!• Do problems 15-18,21,24,29-31

Closure • Today we learned how to fins real zeros • Next class we are going to learn about the fundamental

theorem of algebra.