Factor Trinomials

Grade 9 · algebra · 86 practice problems · read aloud

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Factoring Trinomials

What is Factoring? 🤔

Factoring a trinomial is like "un-multiplying" it. You're rewriting an expression like x² + 5x + 6 as a product of two simpler expressions, like (x + 2)(x + 3). This is super useful for solving quadratic equations and simplifying complex algebraic fractions.

How to Factor Trinomials (a = 1)

When the trinomial is in the form x² + bx + c, follow these steps:

  1. Identify b and c: In x² + bx + c, find the values of b and c.
  2. Find Factor Pairs: Find two numbers that multiply to give c and add to give b.
  3. Write the Factors: Use the two numbers you found to write the answer as (x + first number)(x + second number).

Worked Examples

Example 1: Factor x² + 7x + 12

  1. b = 7, c = 12
  2. Find two numbers that multiply to 12 and add to 7. (3 and 4 work because 3 × 4 = 12 and 3 + 4 = 7).
  3. Answer: (x + 3)(x + 4)

Example 2: Factor x² - 3x - 10

  1. b = -3, c = -10
  2. Find two numbers that multiply to -10 and add to -3. (2 and -5 work because 2 × -5 = -10 and 2 + (-5) = -3).
  3. Answer: (x + 2)(x - 5)

Common Mistakes to Avoid 🚫

  • Wrong Signs: The most common error! Double-check that your numbers multiply to c AND add to b, paying close attention to negative signs.
  • Forgetting the x: The factored form is (x + m)(x + n), not just (m)(n).
  • Confusing Sum and Product: Remember, they MULTIPLY to 'c' and ADD to 'b'. Don't mix these up!

Tips & Tricks

  • Memory Aid: Think "M" for Multiply and "A" for Add. You need the M&A pair for 'c' and 'b'.
  • Check Your Work! You can always FOIL your answer to see if you get back the original trinomial.
  • If 'c' is positive, both of your numbers will have the same sign (both positive if 'b' is positive, both negative if 'b' is negative).

How to Practice

Start simple and build up your skills!

  • Begin with trinomials where all terms are positive (e.g., x² + 6x + 8).
  • Move on to trinomials where 'b' or 'c' is negative.
  • Create your own problems and solve them, or use online practice websites for endless drills.
  • Time yourself to build speed and confidence once you understand the steps.

Practice problems

6 of the 86, worked through step by step — try them before opening the answer.

1 12x² - 5x - 2 = ?

Hint: For factoring quadratic trinomials using the AC method, first multiply the coefficient of x² by the constant term. Then find two numbers that multiply to this product and add to the coefficient of x. Use these numbers to split the middle term and factor by grouping.

Show the answer

Answer: (4x + 1)(3x - 2)

  1. Multiply a and c: 12 × (-2) = -24
  2. Find two numbers that multiply to -24 and add to -5: -8 and 3
  3. Rewrite the middle term: 12x² - 8x + 3x - 2
  4. Factor by grouping: (12x² - 8x) + (3x - 2) = 4x(3x - 2) + 1(3x - 2)
  5. Factor out the common binomial: (3x - 2)(4x + 1)

The answer is (4x + 1)(3x - 2).

2 12x² - 11x - 15 = ?

Hint: Look for two numbers that multiply to the product of the first and last coefficients and add to the middle coefficient

Show the answer

Answer: (4x + 3)(3x - 5)

  1. Identify a = 12, b = -11, c = -15
  2. Multiply a × c = 12 × (-15) = -180
  3. Find two numbers that multiply to -180 and add to -11: -20 and 9
  4. Rewrite the middle term: 12x² - 20x + 9x - 15
  5. Factor by grouping: (12x² - 20x) + (9x - 15) = 4x(3x - 5) + 3(3x - 5)
  6. Factor out the common binomial: (3x - 5)(4x + 3)

The answer is (4x + 3)(3x - 5).

3 Factor: 6x² + 19x + 10

Hint: For a quadratic trinomial ax² + bx + c, find two numbers that multiply to a×c and add to b. Then rewrite the middle term and factor by grouping.

Show the answer

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

  1. Identify coefficients We have a quadratic in the form ax² + bx + c, where: a = 6 b = 19 c = 10
  2. Multiply a and c a * c = 6 * 10 = 60
  3. Find two numbers that multiply to 60 and add to 19 Let's list factor pairs of 60: 1 and 60 → sum = 61 (no) 2 and 30 → sum = 32 (no) 3 and 20 → sum = 23 (no) 4 and 15 → sum = 19 (yes!) So our numbers are 4 and 15.
  4. Rewrite the middle term using these numbers 6x² + 19x + 10 = 6x² + 4x + 15x + 10
  5. Factor by grouping Group the terms: (6x² + 4x) + (15x + 10) Factor each group: From the first group (6x² + 4x), factor out 2x: 2x(3x + 2) From the second group (15x + 10), factor out 5: 5(3x + 2)
  6. Combine the factors We now have: 2x(3x + 2) + 5(3x + 2) Since (3x + 2) is common to both terms, factor it out: (3x + 2)(2x + 5)
  7. Write the final factored form (2x + 5)(3x + 2) To verify, we can expand: 2x * 3x = 6x² 2x * 2 = 4x 5 * 3x = 15x 5 * 2 = 10 Combine: 6x² + 4x + 15x + 10 = 6x² + 19x + 10 ✓ Final answer: (2x + 5)(3x + 2)

Let's factor the quadratic expression: 6x² + 19x + 10

4 Factor: 12x² - 7x - 10

Hint: Look for two numbers that multiply to the product of the leading coefficient and constant term, and add to the middle coefficient

Show the answer

Answer: (4x - 5)(3x + 2)

  1. Identify a = 12, b = -7, c = -10
  2. Multiply a and c: 12 × (-10) = -120
  3. Find two numbers that multiply to -120 and add to -7: -15 and 8
  4. Rewrite the middle term: 12x² - 15x + 8x - 10
  5. Factor by grouping: (12x² - 15x) + (8x - 10)
  6. Factor out common factors: 3x(4x - 5) + 2(4x - 5)
  7. Factor out the common binomial: (4x - 5)(3x + 2)

The answer is (4x - 5)(3x + 2).

5 Factor: 9x² - 21x + 10

Hint: Think about the AC method: multiply the first and last coefficients, then find two numbers that multiply to that product and add to the middle coefficient.

Show the answer

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

  1. Identify a = 9, b = -21, c = 10.
  2. Multiply a and c: 9 × 10 = 90.
  3. Find two numbers that multiply to 90 and add to -21. Since the product is positive and the sum is negative, both numbers must be negative. The pair is -6 and -15 because (-6) × (-15) = 90 and (-6) + (-15) = -21.
  4. Rewrite the middle term using these numbers: 9x² - 6x - 15x + 10.
  5. Factor by grouping: (9x² - 6x) + (-15x + 10).
  6. Factor out the greatest common factor from each group: 3x(3x - 2) - 5(3x - 2).
  7. Factor out the common binomial (3x - 2): (3x - 2)(3x - 5).

The answer is (3x - 2)(3x - 5).

6 Factor: 8x² + 14x - 15

Hint: Think about the AC method: multiply the coefficient of x² by the constant term, then find two numbers that multiply to that product and add to the coefficient of x.

Show the answer

Answer: (4x - 3)(2x + 5)

  1. Identify a = 8, b = 14, c = -15.
  2. Multiply a and c: 8 × (-15) = -120.
  3. Find two numbers that multiply to -120 and add to 14. The numbers are 20 and -6 because 20 × (-6) = -120 and 20 + (-6) = 14.
  4. Rewrite the middle term using these numbers: 8x² + 20x - 6x - 15.
  5. Factor by grouping: (8x² + 20x) + (-6x - 15).
  6. Factor out the greatest common factor from each group: 4x(2x + 5) - 3(2x + 5).
  7. Factor out the common binomial (2x + 5): (2x + 5)(4x - 3).

The answer is (4x - 3)(2x + 5).

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