Factoring by Grouping – ASVAB Math Review

1.Factoring by Grouping

Dividing a polynomial into two pairs, factoring out the GCF from each pair, and then factoring out the common binomial.

ax + ay + bx + by = $$a(x + y) + b(x + y) = (a + b)(x + y)$$

Group the molecules that share common factors. After factoring out the common factors (GCFs) of each group, if a common polynomial appears in the parentheses, factor that polynomial out as the final common factor.

Example

Factor by grouping:

$$3x^3 + 6x^2 + 4x + 8$$

Show solution

Step 1: Group into pairs: $$(3x^3 + 6x^2) + (4x + 8)$$.
Step 2: Factor out GCF from each pair:

$$3x^2(x + 2) + 4(x + 2)$$.

Step 3: Factor out the common binomial $$(x + 2)$$:

$$(3x^2 + 4)(x + 2)$$.

=> Conclusion: $$3x^3 + 6x^2 + 4x + 8 = (3x^2 + 4)(x + 2)$$

Practice problems

Problem 1
Factor by grouping:
$$x^3 + 2x^2 + 3x + 6$$
Show solution
Answer: $$(x^2 + 3)(x + 2)$$
- Step 1: Group pairs: $$(x^3 + 2x^2) + (3x + 6)$$.
- Step 2: Factor GCF from each group: $$x^2(x + 2) + 3(x + 2)$$.
- Step 3: Factor out $$(x + 2)$$: $$(x^2 + 3)(x + 2)$$.
=> Conclusion: $$(x^2 + 3)(x + 2)$$.
=> Correct Option: A ($$(x^2 + 3)(x + 2)$$)
Problem 2
Factor by grouping:
$$2x^3 + 4x^2 + 5x + 10$$
Show solution
Answer: $$(2x^2 + 5)(x + 2)$$
- Step 1: Group pairs: $$(2x^3 + 4x^2) + (5x + 10)$$.
- Step 2: Factor GCF: $$2x^2(x + 2) + 5(x + 2)$$.
- Step 3: Factor out $$(x + 2)$$: $$(2x^2 + 5)(x + 2)$$.
=> Conclusion: $$(2x^2 + 5)(x + 2)$$.
=> Correct Option: A ($$(2x^2 + 5)(x + 2)$$)
Problem 3
Factor by grouping:
$$4x^3 + 12x^2 + 3x + 9$$
Show solution
Answer: $$(4x^2 + 3)(x + 3)$$
- Step 1: Group pairs: $$(4x^3 + 12x^2) + (3x + 9)$$.
- Step 2: Factor GCF: $$4x^2(x + 3) + 3(x + 3)$$.
- Step 3: Factor out $$(x + 3)$$: $$(4x^2 + 3)(x + 3)$$.
=> Conclusion: $$(4x^2 + 3)(x + 3)$$.
=> Correct Option: A ($$(4x^2 + 3)(x + 3)$$)

2.Grouping with Negative Signs

Handling terms with negative signs when grouping.

ax - ay - bx + by = $$a(x - y) - b(x - y) = (a - b)(x - y)$$

When grouping numerators with a minus sign in between, be careful when placing the parentheses. If you place the minus sign outside the parentheses, the signs inside the parentheses must be reversed.

Example

Factor by grouping:

$$x^3 - 3x^2 - 4x + 12$$

Show solution

Step 1: Group into pairs (be careful with middle minus sign):

$$(x^3 - 3x^2) - (4x - 12)$$ [Note: +12 becomes -12].

Step 2: Factor out GCF from each group:

$$x^2(x - 3) - 4(x - 3)$$.

Step 3: Factor out $$(x - 3)$$:

$$(x^2 - 4)(x - 3)$$.

Step 4: Fully factor difference of squares $$(x^2 - 4)$$:

$$(x - 2)(x + 2)(x - 3)$$.

=> Conclusion: $$x^3 - 3x^2 - 4x + 12 = (x - 2)(x + 2)(x - 3)$$

Practice problems

Problem 4
Factor by grouping:
$$x^3 - 5x^2 - 2x + 10$$
Show solution
Answer: $$(x^2 - 2)(x - 5)$$
- Step 1: Group pairs: $$(x^3 - 5x^2) - (2x - 10)$$.
- Step 2: Factor GCF: $$x^2(x - 5) - 2(x - 5)$$.
- Step 3: Factor out $$(x - 5)$$: $$(x^2 - 2)(x - 5)$$.
=> Conclusion: $$(x^2 - 2)(x - 5)$$.
=> Correct Option: B ($$(x^2 - 2)(x - 5)$$)
Problem 5
Factor by grouping:
$$2x^3 - 6x^2 - 3x + 9$$
Show solution
Answer: $$(2x^2 - 3)(x - 3)$$
- Step 1: Group pairs: $$(2x^3 - 6x^2) - (3x - 9)$$.
- Step 2: Factor GCF: $$2x^2(x - 3) - 3(x - 3)$$.
- Step 3: Factor out $$(x - 3)$$: $$(2x^2 - 3)(x - 3)$$.
=> Conclusion: $$(2x^2 - 3)(x - 3)$$.
=> Correct Option: B ($$(2x^2 - 3)(x - 3)$$)
Problem 6
Factor by grouping:
$$x^3 + 4x^2 - 9x - 36$$
Show solution
Answer: $$(x - 3)(x + 3)(x + 4)$$
- Step 1: Group pairs: $$(x^3 + 4x^2) - (9x + 36)$$.
- Step 2: Factor GCF: $$x^2(x + 4) - 9(x + 4)$$.
- Step 3: Factor out $$(x + 4)$$: $$(x^2 - 9)(x + 4)$$.
- Step 4: Factor difference of squares: $$(x - 3)(x + 3)(x + 4)$$.
=> Conclusion: $$(x - 3)(x + 3)(x + 4)$$.
=> Correct Option: B ($$(x - 3)(x + 3)(x + 4)$$)