Factor Polynomials

Grade 9 · algebra · 100 practice problems · read aloud

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Factoring Polynomials

What is Factoring? 🤔

Factoring is like "un-multiplying" a polynomial. You break it down into simpler expressions (called factors) that multiply together to give you the original polynomial. It's super useful for solving quadratic equations and simplifying complicated algebraic expressions.

How to Factor (Step-by-Step)

For a quadratic in the form ax² + bx + c:

  1. Find two numbers that MULTIPLY to a×c and ADD to b.
  2. Rewrite the middle term (bx) using those two numbers.
  3. Factor by grouping the first two terms and the last two terms.
  4. Factor out the common binomial to get your final answer.

Worked Examples

Example 1: Factor x² + 5x + 6

We need two numbers that multiply to 6 and add to 5.

✅ The numbers are 2 and 3 (because 2×3=6 and 2+3=5).

So, x² + 5x + 6 = (x + 2)(x + 3).

Example 2: Factor 2x² + 7x + 3

Multiply a×c = 2×3 = 6. We need two numbers that multiply to 6 and add to 7.

✅ The numbers are 6 and 1.

Rewrite: 2x² + 6x + 1x + 3

Group: (2x² + 6x) + (1x + 3) = 2x(x + 3) + 1(x + 3)

Final Answer: (2x + 1)(x + 3)

Common Mistakes to Avoid ⚠️

  • Wrong signs: Always double-check if your numbers add/subtract to give the correct middle term (b), especially with negatives.
  • Forgetting the GCF: Always look for a Greatest Common Factor first! Factor it out before you start.
  • Incorrect grouping: Make sure the binomials you factor out from each group are identical.

Tips & Tricks

  • Master your multiplication tables! Finding the number pairs is much faster if you know them well.
  • Check your work by "FOIL-ing" your final factors to see if you get the original polynomial.
  • Remember the shortcut for simple trinomials where a=1: Just find the two numbers that multiply to c and add to b.

How to Practice

Start simple and build up!

  1. Begin with trinomials where a=1 (e.g., x² + bx + c).
  2. Move to trinomials where a>1 (e.g., 2x² + 5x + 3).
  3. Practice with negative terms (e.g., x² - 4x - 5).
  4. Always start by looking for a GCF to factor out first.

Try factoring these: x² + 8x + 12, 3x² - 14x - 5, 4x² + 12x + 9

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 Factor completely: 8x³ - 27 = ?

Hint: This expression follows the pattern of a difference of cubes. Look for two terms that are perfect cubes.

Show the answer

Answer: (2x - 3)(4x² + 6x + 9)

  1. Identify the expression as a difference of cubes: 8x³ - 27
  2. Recognize that 8x³ = (2x)³ and 27 = 3³
  3. Apply the difference of cubes formula: a³ - b³ = (a - b)(a² + ab + b²)
  4. Substitute a = 2x and b = 3 into the formula
  5. Calculate: (2x - 3)[(2x)² + (2x)(3) + 3²]
  6. Simplify: (2x - 3)(4x² + 6x + 9) The completely factored form is (2x - 3)(4x² + 6x + 9).

2 Factor completely: 16x⁴ - 81 = ?

Hint: This expression is a difference of two perfect squares. Look for a pattern where both terms are perfect squares, and remember that you may need to factor further after the first step.

Show the answer

Answer: (4x² + 9)(2x + 3)(2x - 3)

  1. Recognize that 16x⁴ - 81 is a difference of squares 16x⁴ = (4x²)² and 81 = 9² So 16x⁴ - 81 = (4x²)² - 9²
  2. Apply the difference of squares formula: a² - b² = (a + b)(a - b) (4x² + 9)(4x² - 9)
  3. Notice that 4x² - 9 is also a difference of squares 4x² = (2x)² and 9 = 3² So 4x² - 9 = (2x)² - 3² = (2x + 3)(2x - 3)
  4. The factor 4x² + 9 cannot be factored further over the real numbers
  5. Write the complete factorization (4x² + 9)(2x + 3)(2x - 3)

The answer is (4x² + 9)(2x + 3)(2x - 3).

3 Factor completely: 6x² - 11x - 10

Hint: Look for two numbers whose product equals the product of the leading coefficient and constant term, and whose sum equals the middle coefficient. Then use grouping to factor.

Show the answer

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

  1. Identify coefficients. We have a = 6, b = -11, c = -10.
  2. Multiply a and c. a * c = 6 * (-10) = -60.
  3. Find two numbers that multiply to -60 and add to b (-11). List factor pairs of -60: 1 and -60 → 1 + (-60) = -59 (no) -1 and 60 → -1 + 60 = 59 (no) 2 and -30 → 2 + (-30) = -28 (no) -2 and 30 → -2 + 30 = 28 (no) 3 and -20 → 3 + (-20) = -17 (no) -3 and 20 → -3 + 20 = 17 (no) 4 and -15 → 4 + (-15) = -11 (yes!) So the numbers are 4 and -15.
  4. Rewrite the middle term (-11x) using these numbers. 6x² - 11x - 10 = 6x² + 4x - 15x - 10.
  5. Factor by grouping. Group the first two terms and the last two terms: (6x² + 4x) + (-15x - 10) Factor out the greatest common factor from each group: From (6x² + 4x), factor out 2x: 2x(3x + 2) From (-15x - 10), factor out -5: -5(3x + 2) Now we have: 2x(3x + 2) - 5(3x + 2)
  6. Factor out the common binomial factor (3x + 2). (3x + 2)(2x - 5)
  7. Write the final factored form. (2x - 5)(3x + 2) Final answer: (2x - 5)(3x + 2)

Let's factor 6x² - 11x - 10 completely.

4 Factor completely: 12x³ - 27x = ?

Hint: Look for the greatest common factor first, then check if what remains is a difference of squares

Show the answer

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

  1. Identify the greatest common factor of 12x³ and 27x GCF = 3x
  2. Factor out 3x: 3x(4x² - 9)
  3. Recognize that (4x² - 9) is a difference of squares: (2x)² - 3²
  4. Factor the difference of squares: (2x - 3)(2x + 3)
  5. Combine all factors: 3x(2x - 3)(2x + 3) The completely factored form is 3x(2x - 3)(2x + 3)

5 Factor completely: 18x³ - 50x = ?

Hint: Look for a greatest common factor first, then check if the remaining expression is a difference of squares

Show the answer

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

  1. Identify the greatest common factor of 18x³ and 50x GCF = 2x
  2. Factor out 2x: 2x(9x² - 25)
  3. Recognize that 9x² - 25 is a difference of squares: (3x)² - (5)²
  4. Apply the difference of squares formula: a² - b² = (a - b)(a + b)
  5. Factor completely: 2x(3x - 5)(3x + 5)

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

6 Factor completely: 6x³ - 24x² - 30x

Hint: First identify the greatest common factor of all terms. Then factor the remaining quadratic expression by finding two numbers that multiply to the constant term and add to the coefficient of the middle term.

Show the answer

Answer: 6x(x - 5)(x + 1)

  1. Look for a greatest common factor (GCF)** All terms have a common factor of 6x: 6x³ ÷ 6x = x² -24x² ÷ 6x = -4x -30x ÷ 6x = -5 So: 6x³ - 24x² - 30x = 6x(x² - 4x - 5) --- **
  2. Factor the quadratic trinomial inside** We now factor x² - 4x - 5. We look for two numbers that multiply to -5 and add to -4. Possible pairs for -5: 1 and -5 → 1 + (-5) = -4 ✓ -1 and 5 → -1 + 5 = 4 ✗ So the correct pair is 1 and -5. Thus: x² - 4x - 5 = (x - 5)(x + 1) --- **
  3. Write the complete factored form** Putting it all together: 6x³ - 24x² - 30x = 6x(x - 5)(x + 1) --- **Final Answer:** 6x(x - 5)(x + 1)

Let's factor the expression step by step. We start with: 6x³ - 24x² - 30x --- **

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