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3-5
Finding 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.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Vocabulary
multiplicity
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
In Lesson 6-4, 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 from Lesson
5-3. 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
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 1a
Solve the polynomial equation by factoring.
2x6 – 10x5 – 12x4 = 0
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 1b
Solve the polynomial equation by factoring.
x3 – 2x2 – 25x = –50
Holt McDougal Algebra 2
3-5
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
3-5
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
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 2a
Identify the roots of each equation. State the
multiplicity of each root.
x4 – 8x3 + 24x2 – 32x + 16 = 0
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 2b
Identify the roots of each equation. State the
multiplicity of each root.
2x6 – 22x5 + 48x4 + 72x3 = 0
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Not all polynomials are factorable, but the Rational Root
Theorem can help you find all possible rational roots of
a polynomial equation.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 3
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?
Step 1 Write an equation to model the volume of
the box.
Let x represent the length in feet. Then the width is
x – 1, and the height is x + 4.
V = lwh.
Multiply the left side.
Set the equation equal to 0.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 3 Continued
Step 2 Use the Rational Root Theorem to identify all
possible rational roots.
Step 3 Test the possible roots to find one that is actually a
root. The width must be positive, so try only positive rational
roots.
p
q
1
2
3
4
Holt McDougal Algebra 2
1
–4
3 –12
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 3 Continued
Step 4 Factor the polynomial. The synthetic
substitution of 2 results in a remainder of 0, so 2
is a root and the polynomial in factored form is
(x – 2)(x2 + 5x + 6).
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Polynomial equations may also have irrational roots.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
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.
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.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 4
Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0.
p = –4 and q = 2
Step 2 Graph y = 2x3 – 3x2 –10x – 4 to find the xintercepts.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 4 Continued
1
Step 3 Test the possible rational root –
.
2
2 –3 –10 –4
1
2
Step 4 Solve 2x2 – 4x – 8 = 0 to find the
remaining roots.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 4 Continued
The fully factored equation is

1
2 x +   x –
2 

( 1+ 5 )  x – (1– 5)
1
The roots are – , 1 + 5 , and 1 - 5 .
2
Holt McDougal Algebra 2