ASVAB Study Guide
Algebra · Lesson 9 of 13

Factoring Trinomials (Leading Coefficient is One)

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

1.Factoring: x²+bx+c

Finding two numbers that multiply to c and add up to b to rewrite the expression.

$$x^2 + bx + c = (x + m)(x + n)$$

Find m,n such that:

1) $$m \cdot n = c$$

2) $$m + n = b$$

Note: If c is positive, m and n have the same sign. If c is negative, they have different signs.

Example

Factor the trinomial:

$$x^2 + 7x + 12$$

Show solution
  • Step 1: Identify $$b = 7$$ and $$c = 12$$ (since c > 0, m and n have the same sign $$(+, +)$$).
  • Step 2: Find factors of 12 that add up to 7: $$3 \times 4 = 12$$ and $$3 + 4 = 7$$.
  • Step 3: Write the factored form: $$(x + 3)(x + 4)$$.

=> Conclusion: $$x^2 + 7x + 12 = (x + 3)(x + 4)$$

Practice problems

  1. Problem 1

    Factor the trinomial:

    $$x^2 + 5x + 6$$

    Show solution

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

    • Step 1: Find two numbers that multiply to 6 and add up to 5.
    • Step 2: Factors of 6: $$2 \times 3 = 6$$, and $$2 + 3 = 5$$.
    • Step 3: Write in factored form: $$(x + 2)(x + 3)$$.

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

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

  2. Problem 2

    Factor the trinomial:

    $$x^2 + 9x + 20$$

    Show solution

    Answer: $$(x + 4)(x + 5)$$

    • Step 1: Find two numbers that multiply to 20 and add up to 9.
    • Step 2: Factors of 20: $$4 \times 5 = 20$$, and $$4 + 5 = 9$$.
    • Step 3: Write in factored form: $$(x + 4)(x + 5)$$.

    => Conclusion: $$(x + 4)(x + 5)$$.

    => Correct Option: A ($$(x + 4)(x + 5)$$)

  3. Problem 3

    Factor the trinomial:

    $$x^2 + 13x + 42$$

    Show solution

    Answer: $$(x + 6)(x + 7)$$

    • Step 1: Find two numbers that multiply to 42 and add up to 13.
    • Step 2: Factors of 42: $$6 \times 7 = 42$$, and $$6 + 7 = 13$$.
    • Step 3: Write in factored form: $$(x + 6)(x + 7)$$.

    => Conclusion: $$(x + 6)(x + 7)$$.

    => Correct Option: B ($$(x + 6)(x + 7)$$)

Remember

Find m, n such that:

1) $$m \cdot n = c$$

2) $$m + n = b$$

Note: If c is positive, m and n have the same sign. If c is negative, they have different signs.

2.Factoring with Negative Signs

Handling cases where b or c are negative numbers.

$$x^2 - bx + c = (x - m)(x - n)$$

$$x^2 \pm bx - c = (x + m)(x - n)$$

If c is $$(+)$$, factors have the same sign as b.If c is $$(-)$$, the larger factor takes the sign of b.

Example

Factor the trinomial:

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

Show solution
  • Step 1: Identify $$b = -2$$ and $$c = -15$$ (since c < 0, factors have opposite signs).
  • Step 2: Find factors of -15 that add up to -2. Since $$b = -2$$, the larger factor must be negative: 3 and $$-5 (3 \times (-5) = -15, 3 + (-5) = -2)$$.
  • Step 3: Write factored form: $$(x - 5)(x + 3)$$.

=> Conclusion: $$x^2 - 2x - 15 = (x - 5)(x + 3)$$

Practice problems

  1. Problem 4

    Factor the trinomial:

    $$x^2 - 7x + 10$$

    Show solution

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

    • Step 1: Since $$c = 10 (+)$$, both factors have the same sign as $$b = -7 (-)$$.
    • Step 2: Find factors of 10 that add to -7: $$(-2) \times (-5) = 10$$, $$(-2) + (-5) = -7$$.
    • Step 3: Factored form: $$(x - 2)(x - 5)$$.

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

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

  2. Problem 5

    Factor the trinomial:

    $$x^2 - 3x - 18$$

    Show solution

    Answer: $$(x - 6)(x + 3)$$

    • Step 1: Since $$c = -18 (-)$$, factors have opposite signs. Since $$b = -3$$, larger factor is negative.
    • Step 2: Factors of -18: $$(-6) \times 3 = -18$$, $$(-6) + 3 = -3$$.
    • Step 3: Factored form: $$(x - 6)(x + 3)$$.

    => Conclusion: $$(x - 6)(x + 3)$$.

    => Correct Option: B ($$(x - 6)(x + 3)$$)

  3. Problem 6

    Factor the trinomial:

    $$x^2 + 4x - 32$$

    Show solution

    Answer: $$(x + 8)(x - 4)$$

    • Step 1: Since $$c = -32 (-)$$, factors have opposite signs. Since $$b = +4$$, larger factor is positive.
    • Step 2: Factors of -32: $$8 \times (-4) = -32$$, $$8 + (-4) = +4$$.
    • Step 3: Factored form: $$(x + 8)(x - 4)$$.

    => Conclusion: $$(x + 8)(x - 4)$$.

    => Correct Option: B ($$(x + 8)(x - 4)$$)

Remember
  • If c is $$(+)$$, factors have the same sign as b (both negative if b is negative).
  • If c is $$(-)$$, factors have opposite signs, and the larger factor takes the sign of b.