Greatest Common Factor and Factoring by Grouping

(Review) Factoring
Definition: A factor is a number, variable, monomial, or polynomial which is multiplies by
another number, variable, monomial, or polynomial to obtain a product.

1. List all the possible factors of the fol lowing numbers :
a. 12
b. 32
c. 19
d. -45
In the above, the number 19 is an example of a ____________ number because its only positive
factors are one and itself .

(Review) Greatest Common Factor
Definition: The greatest common factor of two or more numbers is the largest number that
divides (goes into) the given numbers with a remainder of zero .

2. Find the GCF (greatest common factor) of the following numbers:
a. 36 and 54
b. 15 and 60
c. 21, 42, and 63
d. 28 and 39

Greatest Common Factor of Polynomials
In order to find the GCF of two or more monomials ,
I. Find the GCF of the coefficients;
II. Find the GCF of the variables;
III. Rewrite the GCF as a product of the GCF of the coefficients times the GCF of the variables.

Examples:
1. Find the GCF of x⁴ and x⁷.

Step I : The only coefficients are 1’s, so this is the GCF of the coefficients.

Step II: Rewrite the two monomials as products of xs without using exponents :
x⁴ = x · x · x · x
x⁷ = x · x · x · x · x · x · x
Since each monomial has 4 xs in it, the GCF of the variables is x ⁴.
Step III: The GCF of x⁴ and x⁷ is x⁴.

2. Find the GCF of xy² and x⁵y⁴.
Step I: The only coefficients are 1’s, so this is the GCF of the coefficients.

Step II: Rewrite the two monomials as products of xs and ys without using exponents:
xy²= x · y · y
x⁵y⁴ = x · x · x · x · x · y · y · y · y
Since each monomial has 1 x and 2 ys in it, the GCF of the variables is xy².

Step III: The GCF of xy² and x5y4 is xy².

3. Find the GCF of 24x⁴y and 9x⁷y⁴.

Step I: The GCF of the coefficients 24 and 9 is 3.

Step II: Rewrite the two monomials as products of xs and ys without using exponents:
x⁴y = x · x · x · x · y
x⁷y⁴ = x · x · x · x · x · x · x · y · y · y · y
Since each monomial has 4 xs and 1 y in it, the GCF of the variables is x4y.

Step III: The GCF of 24x4y and 9x7y4 is 3x⁴y.

3. Find the GCF of 14xy and 21x⁵y².

4. Find the GCF of 18c⁹ and -6c⁴x.

5. Find the GCF of 23a⁶b³ and 42a³c².

Factoring
To factor a polynomial, an attempt should be made to find the GCF of the monomials in the
polynomial. Then this GCF should be factored out of the polynomial by “undistributing” the
GCF out of all the monomials in the polynomial. Note that if the leading coefficient is negative ,
then the GCF should also be negative.

Examples:
1. Factor 7x + 14y.
Step I: Find the GCF of the coefficients. GCF(7, 14) = 7.

Step II: Find the GCF of the variable parts. There is none, so nothing changes .

Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front:

2. Factor 8x³ + 6x².
Step I: Find the GCF of the coefficients. GCF(8, 6) = 2.

Step II: Find the GCF of the variable parts. GCF(x³, x²) = x².

Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front:


3. Factor 18y³(x – 1) + 21y(x – 1)
Step I: Find the GCF of the coefficients: GCF(18, 21) = 3.
Step II: Find the GCF of the variable parts: GCF(y³(x – 1), y(x – 1)) = y(x – 1)
Step III: Divide all the monomials by the GCFs and rewrite with the GCFs out front:

6. Factor 3ab + 7b.

7. Factor -28a³b⁷ – 36a²b⁵.

8. Factor 8a²bc – 12ab²c.

9. Factor 9xy³ + 27xy – x⁵. (Hint: What is the leading coefficient?)

10. Factor (9 – 3x)(27y²) – (9 – 3x)(15y³)

Factoring by Grouping
If a polynomial contains four or more terms, it may be helpful to put the terms into groups of two
and factor out a common factor from each of these groups. This is called grouping.

Examples:
1. Factor x²y + 6x + 3xy² + 18y

Step I: Group the terms so that each group shares a common factor:
x²y + 6x + 3xy² + 18y = (x²y + 6x) + (3xy² + 18y)

Step II: Factor out the common terms from each group:
(x²y + 6x) = x(xy + 6)
(3xy² + 18y) = 3y(xy + 6)

Step III: Rewrite the polynomial as the sum of the factored groups:
x²y + 6x + 3xy² + 18y = x(xy + 6) + 3y(xy + 6)
Step IV: Factor the resulting polynomial from Step III:
x(xy + 6) + 3y(xy + 6) = (xy + 6)(x + 3y)

2. Factor 2x³ + 3x² + 2x + 3

Step I: Group the terms so that each group shares a common factor:
2x3 + 3x² + 2x + 3 = (2x³ + 3x²) + (2x + 3)

Step II: Factor out the common terms from each group:
(2x³ + 3x²) = x²(2x + 3)
(2x + 3) = 1(2x + 3) (Note: There’s no common term, so the GCD is 1.)

Step III: Rewrite the polynomial as the sum of the factored groups from Step II.
2x³ + 3x² + 2x + 3 = x²(2x + 3) + 1(2x + 3)

Step IV: Factor out the resulting polynomial from Step III:
x²(2x + 3) + 1(2x + 3) = (x² + 1)(2x + 3)

3. Factor 3x³ + 3x² – 4x – 4.

Step I: Group the terms so that each group shares a common factor:€
3x³+ 3x² – 4x – 4 = (3x³ + 3x²) + (-4x – 4)

Step II: Factor out the common terms from each group:
(3x³ + 3x²) = 3x²(x + 1)
(-4x – 4) = -4(x + 1)

Step III: Rewrite the polynomial as the sum of the factored groups from Step II.
3x³ + 3x² – 4x – 4 = 3x²(x + 1) + (-4(x + 1)) = 3x²(x + 1) – 4(x + 1)

Step IV: Factor out the resulting polynomial from Step III:
3x²(x + 1) – 4(x + 1) = (3x² – 4)(x + 1)

11. Factor 4a³ + 8a + 3a²b² + 6b².

12. Factor 3a²x + a²y – 12x – 4y.

13. 6ax + 6xb + 10ay + 10yb.

14. 6x + 9xy + 10y + 15y².