Download 5.3 Factoring the GCF and Factor by Grouping

Math 152 — Rodriguez
Blitzer — 5.3
Greatest Common Factors and Factor By Grouping
I. Factoring
A. Factor: Factors are numbers you can multiply to get another number.
3 ⋅ 4 = 12 3 and 4 are factors of 12
2 ⋅ 6 = 12
2 and 6 are also factors of 12
2 ⋅ 2 ⋅ 3 = 12 this is the ‘prime factorization’ of 12
B. Factoring a poly: Process of writing the polynomial as the product of other
polynomials.
poly as sum = poly as product
4x + 12 = 2 (
)
or
4x + 12 = 4 (
)
C. If a polynomial cannot be factored (using integer coefficients), we say the polynomial is
prime.
D. A polynomial is factored completely if it is written as a product of prime polynomials.
That is, you can’t factor any factor further.
II. Greatest Common Factor
A. The greatest common factor or GCF is the greatest factor (expression) that evenly
divides each term in the poly.
B. Factor the GCF from each poly.
1) 5x 3 y 5 − 15x 2 y 7
2) 12x 4 y − 4x 3 y 2 + 8x 2 y 3
C. Factor the negative of the GCF from the poly.
1) −4x 3 + 8x 2 − 6x
2) −2x 2 + 16x − 4
D. Factor the greatest common binomial from the poly.
1)
x (y − 4) + 6(y − 4)
2)
y ( 2x − 5 ) − ( 2x − 5 )
3) 15x 6 y 4 − 5x 4 y 3 + 10x 5 y 2
III. Factor by Grouping
Factor by grouping is used when a polynomial has four terms.
Steps to factor by grouping:
1. Group the terms that have a common factor; group them in pairs.
2. Factor out the GCF from each group.
3. If the binomial that remains inside the parenthesis is the same, proceed to factor
out the greatest common binomial. (If not, check your work. If work is correct
and binomial inside the parenthesis is not the same, try re-arranging the terms
and re-grouping.)
Examples:
1) xy − 6x + 4y − 24
2)
3x 3 − 2x 2 − 6x + 4
3)
3x 2 − 6xy + 5xy − 10y 2
4)
x 2 − ax − 5x + 5a
5)
Blitzer — 5.3
x 3 − 3x 2 + 4x −12
Page 2 of 2