Download 3-5 3-5 Finding Real Roots of Polynomial Equations

Finding
Real
Roots
of of
Finding
Real
Roots
3-5
3-5 Polynomial Equations
Polynomial Equations
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 2Algebra
Algebra22
Holt
McDougal
3-5
Finding Real Roots of
Polynomial Equations
Warm Up
Factor completely.
1. 2y3 + 4y2 – 30y 2y(y – 3)(y + 5)
2. 3x4 – 6x2 – 24
3(x – 2)(x + 2)(x2 + 2)
Solve each equation.
3. x2 – 9 = 0
x = – 3, 3
4. x3 + 3x2 – 4x = 0
Holt McDougal Algebra 2
x = –4, 0, 1
3-5
Finding Real Roots of
Polynomial Equations
Essential Question
How can you find all the possible real
zeros of f(x)?
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 3-4, 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
2-3. 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 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
3-5
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.
x = 0, x = –3, x = 2 Solve for x.
The roots are 0, –3, and 2 where 0 has a multiplicity
of 4. The number of roots equals the degree of the
polynomial.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Example 1A Continued
Check Use a graph. The
roots appear to be
located at x = 0, x = –3,
and x = 2. 
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
Solving Equations Using Quadratic Techniques
An expression that is quadratic in form can be written
as au² + bu + c for any numbers a, b, and c, a≠0,
where u is some expression in x. The expression au²
+ bu + c is called the quadratic form of the original
expression. This method of substituting u for x² is
called u substitution.
For example, let u = x² the expression x⁴ + 13x² + 36
can be written as (x²)² + 13(x²) + 36
or u² +13u + 36. Now you can easily factor into
(u+9)(u+4). Substitute u = x² to get (x²+9)(x²+4).
Holt McDougal Algebra 2
3-5
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
Set the equation equal to 0.
(x²)² = x⁴
(x²)² - 26(x²) + 25=0
Factor the trinomial in
(x2 – 25)(x2 – 1) = 0
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.
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 4x6 + 4x5 – 24x4 = 0, the
polynomial has a multiple roots, 0, –3, and 2.
For example, the root 0 is a factor four times
because 4x4 = 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
For 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.
Holt McDougal Algebra 2
3-5
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
3-5
Finding Real Roots of
Polynomial Equations
Example 2B: Identifying Multiplicity
Identify the roots of each equation. State the
multiplicity of each root.
x4 + 8x3 + 18x2 – 27 = 0
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.
Check Use a graph. A
calculator graph shows
a bend near (–3, 0) and
crosses at (1, 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
Step 1 x3 + 3x2 – 4x – 12 = 0 Set the equation equal to 0.
Step 2 Use the Rational Root Theorem to identify all
possible rational roots.
Factors of –12: ±1, ±2, ±3, ±4, ±6, ±12
Step 3 Test the possible roots to find one that is
actually a root.
Use a synthetic substitution table
to organize your work. The first
row represents the coefficients of
the polynomial. The first column
represents the possible roots and
the last column represents the
remainders. Test possibilities to
identify at least one root.
Holt McDougal Algebra 2
3-5
Finding Real Roots of
Polynomial Equations
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).
(x – 2)(x2 + 5x + 6) = 0
Set the equation equal to 0.
(x – 2)(x + 2)(x + 3) = 0
Factor x2 + 5x + 6.
x = 2, x = –2, or x = –3
Set each factor equal to 0,
and solve.
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
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
3-5
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
3-5
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
3-5
Finding Real Roots of
Polynomial Equations
Check It Out! Example 4
Identify all the real roots of 2x3 – 3x2 –10x – 4 = 0.
Step 1 Use the Rational Root Theorem to identify
possible rational roots.
±1, ±2, ±4 = ±1, ±2, ±4, ± 1 .
±1, ±2
2 p = –4 and q = 2
Step 2 Graph y = 2x3 – 3x2 –10x – 4 to find the xintercepts.
The x-intercepts are located at
or near –0.5, –1.2, and 3.2.
The x-intercepts –1.2 and 3.2
do not correspond to any of the
possible rational roots.
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
1
1
Test
–
. The remainder
2 –3 –10 –4
–
2
2
1 ) is a factor.
is
0,
so
(x
+
–1 2 4
2
2 –4 –8 0
1
The polynomial factors into (x + )(2x2 – 4x – 8).
2
Step 4 Solve 2x2 – 4x – 8 = 0 to find the
remaining roots.
2(x2 – 2x – 4) = 0
Factor out the GCF, 2
Use the quadratic formula to
2± 4+16
x=
=1± 5
identify the irrational roots.
2
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
3-5
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
3-5
Finding Real Roots of
Polynomial Equations
Why is the Rational Root Theorem so polite?
It minds its p’s and q’s!
Holt McDougal Algebra 2