Transcript Lesson 3.5A
Factoring Polynomials
•
How do we use the Factor Theorem to determine factors of a polynomial?
•
How do we factor the sum and difference of two cubes Holt McDougal Algebra 2
Factoring Polynomials
Recall that if a number is divided by any of its factors, the remainder is 0. Likewise, if a polynomial is divided by any of its factors, the remainder is 0. The Remainder Theorem states that if a polynomial is divided by ( function at
a x
–
a
), the remainder is the value of the . So, if (
x
–
a
) is a factor of
P
(
x
), then
P
(
a
) = 0.
Holt McDougal Algebra 2
Factoring Polynomials Example 1: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of the polynomial P(x). (x + 1); (x 2 – 3x + 1)
Find
P
(–1) by synthetic substitution.
1 3 1 4 4 5
P
(–1) = 5
P
(–1) ≠ 0, so (
x P
(
x
) =
x
2 – 3
x
+ 1) + 1. is not a factor of
Holt McDougal Algebra 2
Factoring Polynomials Example 2: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of the polynomial P(x). (x + 2); (3x 4 + 6x 3 – 5x – 10)
Find
P
(–2) by synthetic substitution.
2 6 0 5 1 0 6 10 5 0
P
(–2) = 0, so (
x
+ 2) is a factor of
P
(
x
) = 3
x
4 + 6
x
3 – 5
x
– 10.
Holt McDougal Algebra 2
Factoring Polynomials Example 3: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of the polynomial P(x). (x + 2); (4x 2 – 2x + 5)
Find
P
(–2) by synthetic substitution.
2 2 5 8 25
P
(–2) = 25
P
(–2) ≠ 0, so (
x P
(
x
) = 4
x
2 – 2
x
+ 2) + 5. is not a factor of
Holt McDougal Algebra 2
Factoring Polynomials Example 4: Determining Whether a Linear Binomial is a Factor Determine whether the given binomial is a factor of the polynomial P(x).
Divide everything by 3
(3x – 6); (3x 4 (x – 2); (x 4 – 6x – 2x 3 3 + 6x + 2x 2 2 + 3x – 30) + x – 10)
Find
P
(2) by synthetic substitution.
2 2 2 1 0 2 4 2 10 5 0
P
(–2) = 0, so (
x
+ 2) is a factor of
P
(
x
) = 3
x
4 + 6
x
3 – 5
x
– 10.
Holt McDougal Algebra 2
Factoring Polynomials
Methods of Factoring
Binomial
GCF Difference of Two Squares
Trinomial
GCF Perfect Square Trinomial Sum of Two Cubes Two Binomials a = 1 (Shortcut)
4 or more terms
GCF Grouping Difference of Two Cubes Two Binomials a ≠ 1 (Cross)
Holt McDougal Algebra 2
Factoring Polynomials
Lesson 3.5 Practice A
Holt McDougal Algebra 2