Download Factor and Solve Polynomial Equations

5.4
Factor and Solve Polynomial
Equations
Goal
Your Notes
p Factor and solve other polynomial equations.
VOCABULARY
Factored completely
Factor by grouping
Quadratic form
FACTORING POLYNOMIALS
Examples
and
are
.
is not
can be factored as
because
.
(x 1 2)(x 2 2 5x 1 6) is not factored completely
because x 2 2 5x 1 6 5
.
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Lesson 5.4 • Algebra 2 Notetaking Guide
137
5.4
Factor and Solve Polynomial
Equations
Goal
Your Notes
p Factor and solve other polynomial equations.
VOCABULARY
Factored completely A factorable polynomial
with integer coefficients written as a product of
unfactorable polynomials with integer coefficients
Factor by grouping A method used to factor some
polynomials with pairs of terms that have a
common monomial factor
Quadratic form An expression of the form
au2 1 bu 1 c, where u is any expression in x
FACTORING POLYNOMIALS
Examples 5 1 x 2 3 2 1 x 1 2 2 and 2x 1 x2 2 6 2 are
factored completely.
7x2 1 x2 2 9 2 is not factored completely because
1 x2 2 9 2 can be factored as 1 x 1 3 2 1 x 2 3 2.
(x 1 2)(x 2 2 5x 1 6) is not factored completely
because x 2 2 5x 1 6 5 (x 2 2)(x 2 3) .
Copyright © Holt McDougal. All rights reserved.
Lesson 5.4 • Algebra 2 Notetaking Guide
137
Your Notes
SPECIAL FACTORING PATTERNS
Sum of Two Cubes
a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)
Example
x3 1 8 5 (x 1 2)(
)
Difference of Two Cubes
a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2)
Example
8x 3 2 1 5 (2x 2 1)(
Example 1
)
Factor the sum or difference of two cubes
Factor the polynomial completely.
Difference of
two cubes
a. z 3 2 125 5 z 3 2
5 (z 2
)(
)
b. 81y 4 1 192y 5 3y (
Factor common
monomial.
)
5 3y[
1
5 3y(
)(
]
Sum of two
cubes
)
Checkpoint Factor the polynomial completely.
1. 8x 3 1 64
Example 2
Factor by grouping
Factor the polynomial x 3 2 2x 2 2 9x 1 18 completely.
x 3 2 2x 2 2 9x 1 18
) 2 9(
5 x 2(
138
)
Factor by grouping.
5
Distributive property
5
Difference of two
squares
Lesson 5.4 • Algebra 2 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
SPECIAL FACTORING PATTERNS
Sum of Two Cubes
a3 1 b3 5 (a 1 b)(a2 2 ab 1 b2)
Example
x3 1 8 5 (x 1 2)( x 2 2 2x 1 4 )
Difference of Two Cubes
a3 2 b3 5 (a 2 b)(a2 1 ab 1 b2)
Example
8x 3 2 1 5 (2x 2 1)( 4x 2 1 2x 1 1 )
Example 1
Factor the sum or difference of two cubes
Factor the polynomial completely.
Difference of
two cubes
a. z 3 2 125 5 z 3 2 53
5 (z 2 5 )( z 2 1 5z 1 25 )
Factor common
monomial.
b. 81y 4 1 192y 5 3y ( 27y 3 1 64 )
5 3y[ (3y) 3 1 4 3 ]
Sum of two
cubes
5 3y( 3y 1 4 )( 9y 2 2 12y 1 16 )
Checkpoint Factor the polynomial completely.
1. 8x 3 1 64
8(x 1 2)(x 2 2 2x 1 4)
Example 2
Factor by grouping
Factor the polynomial x 3 2 2x 2 2 9x 1 18 completely.
x 3 2 2x 2 2 9x 1 18
5 x 2( x 2 2 ) 2 9( x 2 2 )
138
Factor by grouping.
5 (x 2 2 9)(x 2 2)
Distributive property
5 (x 1 3)(x 2 3)(x 2 2)
Difference of two
squares
Lesson 5.4 • Algebra 2 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Factor polynomials in quadratic form
Factor completely: (a) 16x 4 2 256 and
(b) 3y7 2 15y 5 1 18y 3.
a. 16x 4 2 256 5 (
)2 2
2
5
5
b. 3y7 2 15y 5 1 18y 3 5 3y 3(
)
5
Checkpoint Factor each polynomial completely.
2. x3 1 2x 2 2 25x 2 50
Example 4
3. x 4 2 14x 2 1 45
Solve a polynomial equation
What are the real-number solutions of the equation
x 4 1 9 5 10x 2?
x 4 1 9 5 10x 2 Write original
equation.
x5
,x5
,x5
The solutions are
Homework
50
Write in standard
form.
50
Factor trinomial.
50
Difference of two
squares
Zero product
property
,x5
.
Checkpoint Find the real-number solutions.
4. 2x5 1 24x 5 14x3
Copyright © Holt McDougal. All rights reserved.
Lesson 5.4 • Algebra 2 Notetaking Guide
139
Your Notes
Example 3
Factor polynomials in quadratic form
Factor completely: (a) 16x 4 2 256 and
(b) 3y7 2 15y 5 1 18y 3.
a. 16x 4 2 256 5 ( 4x 2 )2 2 16
2
5 (4x 2 1 16)(4x 2 2 16)
5 (4x 2 1 16)(2x 1 4)(2x 2 4)
b. 3y7 2 15y 5 1 18y 3 5 3y 3( y 4 2 5y 2 1 6 )
5 3y 3(y 2 2 3)(y 2 2 2)
Checkpoint Factor each polynomial completely.
2. x3 1 2x 2 2 25x 2 50
(x 1 5)(x 2 5)(x 1 2)
Example 4
3. x 4 2 14x 2 1 45
(x 2 2 5)(x 1 3)(x 2 3)
Solve a polynomial equation
What are the real-number solutions of the equation
x 4 1 9 5 10x 2?
x 4 1 9 5 10x 2 Write original
equation.
x 4 2 10x 2 1 9 5 0
Write in standard
form.
(x 2 2 9)(x 2 2 1) 5 0
Factor trinomial.
(x 1 3)(x 2 3)(x 1 1)(x 2 1) 5 0
Difference of two
squares
x 5 23 , x 5 3 , x 5 21 , x 5 1
Zero product
property
The solutions are 23, 3, 21, and 1 .
Homework
Checkpoint Find the real-number solutions.
4. 2x5 1 24x 5 14x3
}
}
0, 2Ï 3 , Ï3 , 22, 2
Copyright © Holt McDougal. All rights reserved.
Lesson 5.4 • Algebra 2 Notetaking Guide
139