ASVAB Study Guide
Algebra · Lesson 10 of 13

Factoring Trinomials (Leading Coefficient is NOT one)

Key rules and formulas, worked examples and 6 practice problems with step-by-step solutions.

1.The AC Method (Grouping)

Breaking the middle term (b) into two parts to factor by grouping.

Find m, n where:

$$m \cdot n = a \cdot c$$

$$m + n = b$$

Multiply a and c first. Then find factors of that product that add up to b. Use those factors to split bx into mx+nx, then group the terms.

Example

Factor the trinomial:

$$2x^2 + 7x + 3$$

Show solution
  • Step 1: Compute $$a \cdot c = 2 \cdot 3 = 6$$. Find m, n with $$m \cdot n = 6$$ and $$m + n = 7 \Rightarrow m = 1$$, $$n = 6$$.
  • Step 2: Split $$7x$$ into $$6x + 1x$$: $$2x^2 + 6x + x + 3$$.
  • Step 3: Group terms: $$(2x^2 + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3)$$.
  • Step 4: Factor out common factor $$(x + 3)$$: $$(2x + 1)(x + 3)$$.

=> Conclusion: $$2x^2 + 7x + 3 = (2x + 1)(x + 3)$$

Practice problems

  1. Problem 1

    Factor using AC Method:

    $$2x^2 + 5x + 2$$

    Show solution

    Answer: $$(2x + 1)(x + 2)$$

    • Step 1: $$a \cdot c = 2 \cdot 2 = 4$$. Find factors adding to 5: 1 and 4.
    • Step 2: Split $$5x$$: $$2x^2 + 4x + x + 2$$.
    • Step 3: Group: $$2x(x + 2) + 1(x + 2)$$.
    • Step 4: Factored form: $$(2x + 1)(x + 2)$$.

    => Conclusion: $$(2x + 1)(x + 2)$$.

    => Correct Option: A ($$(2x + 1)(x + 2)$$)

  2. Problem 2

    Factor using AC Method:

    $$3x^2 - 5x - 2$$

    Show solution

    Answer: $$(3x + 1)(x - 2)$$

    • Step 1: $$a \cdot c = 3 \cdot (-2) = -6$$. Find factors adding to -5: -6 and 1.
    • Step 2: Split $$-5x$$: $$3x^2 - 6x + x - 2$$.
    • Step 3: Group: $$3x(x - 2) + 1(x - 2)$$.
    • Step 4: Factored form: $$(3x + 1)(x - 2)$$.

    => Conclusion: $$(3x + 1)(x - 2)$$.

    => Correct Option: A ($$(3x + 1)(x - 2)$$)

  3. Problem 3

    Factor using AC Method:

    $$6x^2 + 11x - 10$$

    Show solution

    Answer: $$(2x + 5)(3x - 2)$$

    • Step 1: $$a \cdot c = 6 \cdot (-10) = -60$$. Find factors adding to 11: 15 and -4.
    • Step 2: Split $$11x$$: $$6x^2 + 15x - 4x - 10$$.
    • Step 3: Group: $$3x(2x + 5) - 2(2x + 5)$$.
    • Step 4: Factored form: $$(3x - 2)(2x + 5)$$.

    => Conclusion: $$(2x + 5)(3x - 2)$$.

    => Correct Option: B ($$(2x + 5)(3x - 2)$$)

2.Trial and Error Method

Testing combinations of factors of a and c to get b.

(px + r)(qx + s)

where $$p \cdot q = a$$, $$r \cdot s = c$$

Use the FOIL method in reverse. Identify factors of the first term (a) and last term (c), then test ""Outer + Inner"" products to see if they equal b.

Example

Factor using Trial and Error:

$$3x^2 + 10x + 8$$

Show solution
  • Step 1: Factors of $$a = 3$$ are (1, 3). So structure is $$(3x + r)(x + s)$$.
  • Step 2: Factors of $$c = 8$$ are (1, 8), (2, 4).
  • Step 3: Test 'Outer + Inner':
  • If $$r = 2$$, $$s = 4$$: Outer $$= 3x \cdot 4 = 12x$$, Inner $$= 2 \cdot x = 2x$$ => Total $$= 14x$$ (Incorrect).
  • If $$r = 4$$, $$s = 2$$: Outer $$= 3x \cdot 2 = 6x$$, Inner $$= 4 \cdot x = 4x$$ => Total $$= 10x$$ (Correct!).
  • Step 4: Write result: $$(3x + 4)(x + 2)$$.

=> Conclusion: $$3x^2 + 10x + 8 = (3x + 4)(x + 2)$$

Practice problems

  1. Problem 4

    Factor using Trial & Error:

    $$2x^2 + 7x + 5$$

    Show solution

    Answer: $$(2x + 5)(x + 1)$$

    • Step 1: Factors of 2 are (2, 1). Form: $$(2x + r)(x + s)$$.
    • Step 2: Factors of 5 are (5, 1).
    • Step 3: Test Outer + Inner: $$(2x \cdot 1) + (5 \cdot x) = 2x + 5x = 7x$$.
    • Step 4: Factored form: $$(2x + 5)(x + 1)$$.

    => Conclusion: $$(2x + 5)(x + 1)$$.

    => Correct Option: A ($$(2x + 5)(x + 1)$$)

  2. Problem 5

    Factor using Trial & Error:

    $$5x^2 - 13x + 6$$

    Show solution

    Answer: $$(5x - 3)(x - 2)$$

    • Step 1: Factors of 5 are (5, 1). Since b < 0, c > 0, signs are negative: $$(5x - r)(x - s)$$.
    • Step 2: Factors of 6 are (3, 2).
    • Step 3: Test Outer + Inner: $$(5x \cdot (-2)) + (-3 \cdot x) = -10x - 3x = -13x$$.
    • Step 4: Factored form: $$(5x - 3)(x - 2)$$.

    => Conclusion: $$(5x - 3)(x - 2)$$.

    => Correct Option: B ($$(5x - 3)(x - 2)$$)

  3. Problem 6

    Factor using Trial & Error:

    $$4x^2 - 4x - 15$$

    Show solution

    Answer: $$(2x - 5)(2x + 3)$$

    • Step 1: Factors of 4 can be (2, 2) or (4, 1). Test $$(2, 2)$$: $$(2x + r)(2x + s)$$.
    • Step 2: Factors of -15 are $$(-5, 3)$$.
    • Step 3: Test Outer + Inner: $$(2x \cdot 3) + (-5 \times 2x) = 6x - 10x = -4x$$.
    • Step 4: Factored form: $$(2x - 5)(2x + 3)$$.

    => Conclusion: $$(2x - 5)(2x + 3)$$.

    => Correct Option: B ($$(2x - 5)(2x + 3)$$)