3.06 More Factoring Patterns
Warm – up:
Which expression represents 𝑥 2 − 16𝑥 + 64 in factored form?
a.
b.
c.
d.
e.
(𝑥 + 8)2
(𝑥 + 8)(𝑥 − 8)
(𝑥 − 16)(𝑥 + 4)
(𝑥 − 8)2
I don’t think it can be factored
Today’s Targeted Task
Today I am…
Investigating different factoring methods.
So that I can…
Factor polynomials in the future!
(this is the MOST important thing for me to learn in Algebra IIA)
Let’s Review:
1
2
3
4
Match the polynomial on the left with the correct type of factoring pattern
Polynomial
Factoring Pattern
3
6
A Trinomial with a leading coefficient of 1
8𝑚 + 𝑛
B Difference of cubes
25𝑝2 − 36𝑞 8
2
C Difference of Squares
𝑥 − 9𝑥 + 20
9
3
D Sum of Cubes
64𝑠 − 𝑡
Summary:
Ex. 1) Which expression is the completely factored form of 8𝑚3 + 𝑛6 ?
a)
(2𝑚 + 𝑛²)³
b)
(2𝑚 + 𝑛²)(4𝑚² + 𝑛⁴)
c)
(2𝑚 − 𝑛²)(4𝑚² + 2𝑚𝑛² + 𝑛⁴)
d)
(2𝑚 + 𝑛²)(4𝑚² − 2𝑚𝑛² + 𝑛⁴)
Factor out a GFC, then factor the trinomial
Ex. 2) Factor:
2𝑥 2 + 16𝑥 + 30
Ex. 3) Factor:
5𝑥 2 − 15𝑥 − 50
a. 5(𝑥 + 10)(𝑥 − 1)
b. 5(x − 2)(x + 5)
c. 5(x − 5)(x + 2)
d. 5(x − 10)(x – 1)
e. I am not sure
Ex. 4) Factor: 75𝑦² − 12
Difference of Squares Pattern
𝑎² − 𝑏² = (𝑎 – 𝑏)(𝑎 + 𝑏)
Ex. 5) What is the missing term in the factorization?
50𝑎² − 98 = 2(5𝑎 + __)(5𝑎 – 7)
Factor Out a Common Binomial Factor
It is possible for a greatest common factor (GCF) to be a binomial rather than a monomial. The process of
factoring out a binomial GCF is the same as the process for factoring out a monomial GCF.
Consider these two examples: one has a monomial GCF and one has a binomial GCF.
Example 1:
The GCF of 5𝑥𝑦 − 16𝑦 is a monomial, 𝑦, and the expression is factored as 𝑦(5𝑥 − 16).
Example 2:
The GCF of 5𝑥(𝑥 + 3) − 16(𝑥 + 3) is a binomial, (𝑥 + 3), and the expression is factored as (𝑥 + 3)(5𝑥 − 16).
Ex. 6)
Polynomial
GCF
Factor completely
4𝑥(𝑥 – 1) + 9(𝑥 – 1)
𝑥 − 1
4𝑥(𝑥 – 1) + 9(𝑥 – 1) = (𝑥 – 1)(4𝑥 + 9)
𝑥(𝑥 + 7) – (𝑥 + 7)
𝑥²(𝑥 + 2) – 25(𝑥 + 2)
Ex. 7) Factor.
3𝑚(𝑛 + 2) – 4(𝑛 + 2)
Ex. 8) Factor.
2𝑎(𝑏 − 5) + 3(𝑏 − 5)
Factor by Grouping:
Ex. 9) What type of factoring form is this?
x³ + 2x² - 25x - 50
A Summary of the Steps to Factoring by Grouping:
1. Always look for a GCF first. If there is one, factor it out.
2. Group the four terms in two pairs. (First 2 terms)+(Last 2 terms)
3. Factor a GCF out of each pair.
 If a pair has no GCF, Factor out a 1.
 If one of the leading coefficients is negative, factor out a negative GCF.
4. If Step 2 does not result in two terms with a common binomial factor, rearrange the terms in
the original polynomial and start over.
5. Factor out the common binomial factor
6. Factor completely. Factors with exponents may be able to factor again.
7. Check by multiplying the factors together and comparing the results to the original.
Steps to factor by grouping:
1) Is there a GCF?
2) Group into 2 pairs.
3) Factor a GCF out of each pair. (watch
for negatives)
4) If no GCF, regroup.
5) Factor out the common binomial.
6) Can it be factored further?
7) Check.
Ex. 10) Factor: 𝑥³ + 2𝑥² − 25𝑥 − 50
Ex. 11) Factor: 12𝑎𝑏 + 12𝑎 + 9𝑎² + 16𝑏
Ex. 12) Factor: 18𝑎𝑏 + 15𝑏 − 30𝑎2 − 25𝑎
a) (6𝑎 + 5)(3𝑏 + 5𝑎)
b) (6𝑎 + 5)(3𝑏 – 5𝑎)
c) (6𝑎 – 5)(3𝑏 – 5𝑎)
d) (6𝑎 – 5)(3𝑏 + 5𝑎)
Factoring with a leading coefficient 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄
Steps to factor 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
5𝑥 2 − 12𝑥 + 7
1. Multiply the first and last terms together
5*7 = 35
2. Find factors of new number
1, 35 / -1, -35
5, 7/ -5, -7
3. Pick combination that adds to middle
term
4. Expand the middle term
-5 + -7 = -12
5. Factor by grouping
(5𝑥 2 − 5𝑥) + (−7𝑥 + 7)
5𝑥(𝑥 − 1) − 7(𝑥 − 1)
(𝟓𝒙 − 𝟕)(𝒙 − 𝟏)
6. Check using FOIL
5𝑥 2 − 5𝑥 − 7𝑥 + 7
5𝑥 2 − 12𝑥 + 7
Ex. 13) Factor 2𝑦 2 + 5𝑦 − 7
Ex. 14) Factor 3𝑥 2 − 14𝑥 + 8
-12x = -5x – 7x
5𝑥 2 − 5𝑥 − 7𝑥 + 7
Ex. 15) Factor: 2𝑥 2 + 9𝑥 + 10