warm-up: september 22, 2015 simplify
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Warm-Up: September 22, 2015Simplify
Homework Questions?
Factoring Polynomials
Section P.5
Factoring is the process of writing a polynomial as the product of two or more polynomials.
Prime polynomials cannot be factored using integer coefficients.
Factor completely means keep factoring until everything is prime
Factoring
Greatest common factor Difference of two perfect squares Perfect-square trinomials Factoring x2 + bx + c (big X) Factoring ax2 + bx + c (big X) Factor by grouping – use with 4 terms Sum and difference of perfect cubes
Methods of Factoring
Find the greatest common factor (GCF) of all terms.
Divide each term by the greatest common factor.
Write the GCF outside parenthesis, with the rest of the divided terms added together inside
3a2 – 12a 3a is the GCF
Factoring out greatest common factor
4312,
33 2
aaa
aa
43 aa
a. 18x3 + 27x2 b. x2(x + 3) + 5(x + 3)
You-Try #1: Factor
Works with an even number of terms. Split the terms into two groups. Factor each group separately using GCF. If factor by grouping is possible, the part
inside parentheses of each group will be the same.
Treat the parentheses as common factors to finish factoring.
Factor by Grouping
1535 23 xxxExample 2: Factor
1226 23 xxxYou-Try #2: Factor
Factoring x2 + bx + c Look for integers r and s such that:
◦ r × s = c◦ r + s = b
sxrxcbxx 2
c
b
r s
Example 31272 xx
You-Try #3652 xx
You-Try #3652 xx
Factoring ax2 + bx + c Look for integers r and s such that:
◦ r × s = ac◦ r + s = b
Divide r and s by a, then reduce fractions In your factors, any remaining denominator
gets moved in front of the x
ac
b
r s
Example 46135 2 xx
You-Try #43116 2 xx
Factoring Perfect Square Trinomials
Let A and B be real numbers, variables, or algebraic expressions, 1. A2 + 2AB + B2 = (A + B)2
2. A2 – 2AB + B2 = (A – B)2
Factor: 16x2 – 56x + 49
Example 7
Factor: x2 + 14x + 49
You-Try #7
bababa 22
yxyxyx
xxx
xxx
75754925
121214
339
22
2
2
Difference of Two Perfect Squares
Factoring the Sum and Difference of 2 Cubes
64x3 – 125 = (4x)3 – 53 = (4x – 5)((4x)2 + (4x)(5) + 52) = (4x – 5)(16x2 + 20x + 25)
A3 – B3 = (A – B)(A2 + AB + B2)
x3 + 8 = x3 + 23 = (x + 2)( x2 – x·2 + 22) = (x + 2)( x2 – 2x + 4)
A3 + B3 = (A + B)(A2 – AB + B2)ExampleType
2233 BABABABA
33 8125 yx
Example 8
100027 3 x
You-Try #8
1. If there is a common factor, factor out the GCF.2. Determine the number of terms in the
polynomial and try factoring as follows:a) If there are two terms, can the binomial be factored
by one of the special forms including difference of two squares, sum of two cubes, or difference of two cubes?
b) If there are three terms, is the trinomial a perfect square trinomial? If the trinomial is not a perfect square trinomial, try factoring using the big X.
c) If there are four or more terms, try factoring by grouping.
3. Check to see if any factors with more than one term in the factored polynomial can be factored further. If so, factor completely.
A Strategy for Factoring a Polynomial
Factoring FlowchartFactor out GCF
Count number of terms2
3
4
Factor byGrouping
Check for:• Difference of perfect squares• Sum of perfect cubes• Difference of perfect cubes
1. Check for perfect square trinomial
2. Use big X factoring
Check each factor to see if it can be factored further
Page 53 #1-75 Odd
Assignment
In Exercises 1-10, factor out the greatest common factor.
In Exercises 11-16, factor by grouping.
In Exercises 17-30, factor each trinomial, or state that the trinomial is prime.
158)21
152)19
65)17
4623)15
22)13
1052)11
3123)9
535)727189)5
63)3
2718)1
2
2
2
23
23
23
2
234
2
xx
xx
xx
xxx
xxx
xxx
xxx
xxxxxx
xx
x
In Exercises 17-30, factor each trinomial, or state that the trinomial is prime.
In Exercises 31-40, factor the difference of two squares.
8116)39
16)37
259)35
4936)33
100)31
15164)29
4116)27
28253)25
23)23
4
4
22
2
2
2
2
2
2
x
x
yx
x
x
xx
xx
xx
xx
In Exercises 41-48, factor any perfect square trinomials, or state that the polynomial is prime.
In Exercises 49-56, factor using the formula for the sum or difference of two cubes.
In Exercises 57-84, factor completely, or state that the polynomial is prime.
1622)61
2444)59
33)57
2764)55
18)53
64)51
27)49
169)47
144)45
4914)43
12)41
4
2
3
3
3
3
3
2
2
2
2
x
xx
xx
x
x
x
x
xx
xx
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