Factoring Polynomials

College Algebra - City College of Manila (Philippines)
Notes and Reviewer on Factoring Polynomials

Topic Procedure

Factoring Common Factoring ● Determine the greatest common factor (GCF)
● Rewrite the polynomial as a product of the
greatest common factor and the remaining
factor.
2
Example: 2𝑥 + 10𝑥
𝐺𝐶𝐹 = 2𝑥
Answer: 2x(x + 5)

Factoring by Grouping Example: ay + az + by + bz
= (ay + az) + (by + bz)
Find the GCF each group
GCF of ay and az = a
GCF of by and bz = b
= a(y + z) + b(y + z)
Answer: (y + z)(a + b)

Factoring the Difference of Two Squares Pattern: 𝑎
2
− 𝑏
2
= (𝑎 + 𝑏)(𝑎 − 𝑏)
2
Example: 9𝑥 − 4
Answer: (3x + 2)(3x - 2)

Factoring the Perfect Square Trinomials Patterns:
2 2 2
𝑎 + 2𝑎𝑏 + 𝑏 = (𝑎 + 𝑏)
2 2 2
𝑎 − 2𝑎𝑏 + 𝑏 = (𝑎 − 𝑏)
Note: First term and last term should be perfect square
Examples:
2
1.) 𝑥 + 2𝑥 + 9
Answer: (x + 3)2
2
2.) 25𝑦 − 20𝑦 + 4
Answer: (5y - 2)2

Factoring the Sum of Two Cubes Pattern:
3 3 2 2
𝑎 + 𝑏 = (𝑎 + 𝑏)(𝑎 − 𝑎𝑏 + 𝑏 )

Example:
3 3
8𝑥 + 27𝑦
Factor of first term: 8x3 = (2x)3
Factor of second term: 27y3 = (3y)3

3 3
8𝑥 + 27𝑦 = (2x + 3y)[(2x)2 - (2x)(3y) + (3y)2]
Answer: (2x + 3y)(4x2 - 6xy + 9y2)

Factoring the Difference of Two Cubes Pattern:
3 3 2 2
𝑎 − 𝑏 = (𝑎 − 𝑏)(𝑎 + 𝑎𝑏 + 𝑏 )

Example:
6
125𝑥 + 8