Transcript a2_ch06_05 - crjmathematics

6-5
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
6-5
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
6-5
Finding Real Roots of
Polynomial Equations
Example 1A: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
4x6 + 4x5 – 24x4 = 0
4x4(x2 + x – 6) = 0
Factor out the GCF, 4x4.
4x4(x + 3)(x – 2) = 0
Factor the quadratic.
4x4 = 0 or (x + 3) = 0 or (x – 2) = 0 Set each factor
equal to 0.
Solve for x.
x = 0, x = –3, x = 2
The roots are 0, –3, and 2.
Holt McDougal Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Example 1B: Using Factoring to Solve Polynomial
Equations
Solve the polynomial equation by factoring.
x4 + 25 = 26x2
x4 – 26 x2 + 25 = 0
(x2 – 25)(x2 – 1) = 0
Set the equation equal to 0.
Factor the trinomial in
quadratic form.
(x – 5)(x + 5)(x – 1)(x + 1) Factor the difference of two
squares.
x – 5 = 0, x + 5 = 0, x – 1 = 0, or x + 1 =0
x = 5, x = –5, x = 1 or x = –1
The roots are 5, –5, 1, and –1.
Holt McDougal Algebra 2
Solve for x.
6-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 1b
Solve the polynomial equation by factoring.
x3 – 2x2 – 25x = –50
x3 – 2x2 – 25x + 50 = 0
Set the equation equal to 0.
(x + 5)(x – 2)(x – 5) = 0
Factor.
x + 5 = 0, x – 2 = 0, or x – 5 = 0
x = –5, x = 2, or x = 5
Solve for x.
The roots are –5, 2, and 5.
Holt McDougal Algebra 2
6-5
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
6-5
Finding Real Roots of
Polynomial Equations
Example 2A: Identifying Multiplicity
Identify the roots of each equation. State the
multiplicity of each root.
x3 + 6x2 + 12x + 8 = 0
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.
Check Use a graph. A
calculator graph shows
a bend near (–2, 0). 
Holt McDougal Algebra 2
6-5
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
x4 – 8x3 + 24x2 – 32x + 16 = (x – 2)(x – 2)(x – 2)(x – 2)
x – 2 is a factor four
times. The root 2 has
a multiplicity of 4.
Check Use a graph. A
calculator graph shows
a bend near (2, 0). 
Holt McDougal Algebra 2
6-5
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
6-5
Finding Real Roots of
Polynomial Equations
Polynomial equations may also have irrational roots.
Holt McDougal Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Example 4: Identifying All of the Real Roots of a
Polynomial Equation
Identify all the real roots of 2x3 – 9x2 + 2 = 0.
Step 1 Use the Rational Root Theorem to identify
possible rational roots.
±1, ±2 = ±1, ±2, ± 1 .
p = 2 and q = 2
±1, ±2
2
Step 2 Graph y = 2x3 – 9x2 + 2 to find the x-intercepts.
The x-intercepts are located at
or near –0.45, 0.5, and 4.45.
The x-intercepts –0.45 and
4.45 do not correspond to any
of the possible rational roots.
Holt McDougal Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Example 4 Continued
1
Step 3 Test the possible rational root
.
2
1 . The remainder is
1
Test
2 –9 0 2
2 1
2
0, so (x – ) is a factor.
1 –4 –2
2
2 –8 –4 0
1
The polynomial factors into (x –
)(2x2 – 8x – 4).
2
Step 4 Solve 2x2 – 8x – 4 = 0 to find the
remaining roots.
2(x2 – 4x – 2) = 0
Factor out the GCF, 2
Use the quadratic formula to
4± 16+8
=2  6
x=
identify the irrational roots.
2
Holt McDougal Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Example 4 Continued
The fully factored equation is
(
)
(

1
2  x –   x – 2 + 6  x – 2 –
2

)
6  = 0
1
The roots are
, 2 + 6 , and 2 - 6 .
2
Holt McDougal Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
Lesson Quiz
Solve by factoring.
1. x3 + 9 = x2 + 9x
–3, 3, 1
Identify the roots of each equation. State the
multiplicity of each root.
0 and 2 each with
2. 5x4 – 20x3 + 20x2 = 0
multiplicity 2
3. x3 – 12x2 + 48x – 64 = 0
4 with multiplicity 3
4. A box is 2 inches longer than its height. The width
is 2 inches less than the height. The volume of the
box is 15 cubic inches. How tall is the box? 3 in.
5. Identify all the real roots of x3 + 5x2 – 3x – 3 = 0.
1, -3 + 6, -3 - 6
Holt McDougal Algebra 2