Grade: Grade 9 Subject: Mathematics Unit: Introduction to Quadratics SAT: AdvancedMath ACT: Math

Factoring Basics

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Factoring

Factoring is the process of writing a polynomial as a product of simpler expressions. For quadratic expressions of the form ax² + bx + c, factoring means rewriting it as (px + q)(rx + s) where the product equals the original expression.

Factoring quadratics is one of the most important skills in Algebra I. It allows us to solve quadratic equations, simplify expressions, and understand the structure of polynomial functions.

Why Factor?

  • To solve quadratic equations by setting each factor equal to zero
  • To find x-intercepts (roots) of quadratic functions
  • To simplify rational expressions
  • To identify patterns and relationships in algebra

The Zero Product Property

If the product of two factors equals zero, then at least one of the factors must equal zero. In symbols: If ab = 0, then a = 0 or b = 0. This property is why factoring helps us solve equations.

Types of Factoring Patterns

Pattern Name Form Factored Form Example
Greatest Common Factor (GCF) ax + ay a(x + y) 3x + 6 = 3(x + 2)
Difference of Squares a² - b² (a + b)(a - b) x² - 9 = (x + 3)(x - 3)
Perfect Square Trinomial a² + 2ab + b² (a + b)² x² + 6x + 9 = (x + 3)²
Simple Trinomial (a = 1) x² + bx + c (x + p)(x + q) x² + 5x + 6 = (x + 2)(x + 3)

Factoring x² + bx + c (when a = 1)

To factor a trinomial of the form x² + bx + c:

  1. Find two numbers p and q such that p × q = c (the constant term)
  2. These same numbers must also satisfy p + q = b (the middle coefficient)
  3. Write the factored form as (x + p)(x + q)

Sign Patterns for Factoring x² + bx + c

If c is... If b is... Then p and q are...
Positive Positive Both positive
Positive Negative Both negative
Negative Either One positive, one negative (larger has same sign as b)

SAT/ACT Connection

Factoring questions appear frequently on both the SAT and ACT. You'll need to factor quickly to solve equations and find zeros of functions. Recognizing special patterns like difference of squares can save significant time on test day.

Examples

Work through these examples to master the factoring process.

Example 1: Greatest Common Factor

Problem: Factor 4x² + 12x

Step 1: Identify the GCF of both terms

4x² = 4 · x · x and 12x = 4 · 3 · x

GCF = 4x

Step 2: Factor out the GCF

4x² + 12x = 4x(x + 3)

Check: 4x(x + 3) = 4x² + 12x ✓

Example 2: Difference of Squares

Problem: Factor x² - 25

Step 1: Recognize this as a difference of squares: a² - b²

x² - 25 = x² - 5²

Step 2: Apply the pattern (a + b)(a - b)

x² - 25 = (x + 5)(x - 5)

Check: (x + 5)(x - 5) = x² - 5x + 5x - 25 = x² - 25 ✓

Example 3: Simple Trinomial (positive c, positive b)

Problem: Factor x² + 7x + 12

Step 1: Find two numbers that multiply to 12 and add to 7

Factors of 12: 1×12, 2×6, 3×4

3 + 4 = 7 ✓

Step 2: Write the factored form

x² + 7x + 12 = (x + 3)(x + 4)

Check: (x + 3)(x + 4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓

Example 4: Simple Trinomial (positive c, negative b)

Problem: Factor x² - 8x + 15

Step 1: Find two numbers that multiply to 15 and add to -8

Since c is positive and b is negative, both numbers are negative

-3 × -5 = 15 and -3 + (-5) = -8 ✓

Step 2: Write the factored form

x² - 8x + 15 = (x - 3)(x - 5)

Check: (x - 3)(x - 5) = x² - 5x - 3x + 15 = x² - 8x + 15 ✓

Example 5: Simple Trinomial (negative c)

Problem: Factor x² + 2x - 15

Step 1: Find two numbers that multiply to -15 and add to 2

Since c is negative, one number is positive and one is negative

5 × -3 = -15 and 5 + (-3) = 2 ✓

Step 2: Write the factored form

x² + 2x - 15 = (x + 5)(x - 3)

Check: (x + 5)(x - 3) = x² - 3x + 5x - 15 = x² + 2x - 15 ✓

Practice

Factor each expression completely. Select the correct answer.

1. Factor: 5x + 15

A) 5(x + 3)   B) 5(x + 15)   C) x(5 + 15)   D) Cannot be factored

2. Factor: x² - 16

A) (x - 4)(x - 4)   B) (x + 4)(x + 4)   C) (x + 4)(x - 4)   D) (x - 8)(x + 2)

3. Factor: x² + 9x + 20

A) (x + 4)(x + 5)   B) (x + 2)(x + 10)   C) (x + 1)(x + 20)   D) (x - 4)(x - 5)

4. Factor: x² - 6x + 8

A) (x - 2)(x + 4)   B) (x - 2)(x - 4)   C) (x + 2)(x - 4)   D) (x + 2)(x + 4)

5. Factor: x² - x - 12

A) (x + 3)(x - 4)   B) (x - 3)(x + 4)   C) (x + 3)(x + 4)   D) (x - 3)(x - 4)

6. Factor: x² + 10x + 25

A) (x + 5)(x - 5)   B) (x + 5)²   C) (x - 5)²   D) (x + 25)(x + 1)

7. Factor: x² - 49

A) (x - 7)²   B) (x + 7)²   C) (x + 7)(x - 7)   D) Cannot be factored

8. Factor: x² - 5x - 14

A) (x + 2)(x - 7)   B) (x - 2)(x + 7)   C) (x + 2)(x + 7)   D) (x - 2)(x - 7)

9. Factor: 3x² - 12

A) 3(x² - 4)   B) 3(x + 2)(x - 2)   C) (3x + 6)(x - 2)   D) Both A and B

10. Factor: x² + 4x - 21

A) (x + 7)(x - 3)   B) (x - 7)(x + 3)   C) (x + 7)(x + 3)   D) (x - 7)(x - 3)

Click to reveal answers
  1. A) 5(x + 3) - GCF is 5; 5x + 15 = 5(x + 3)
  2. C) (x + 4)(x - 4) - Difference of squares: x² - 16 = x² - 4²
  3. A) (x + 4)(x + 5) - 4 × 5 = 20 and 4 + 5 = 9
  4. B) (x - 2)(x - 4) - (-2) × (-4) = 8 and -2 + (-4) = -6
  5. A) (x + 3)(x - 4) - 3 × (-4) = -12 and 3 + (-4) = -1
  6. B) (x + 5)² - Perfect square trinomial: (x + 5)(x + 5)
  7. C) (x + 7)(x - 7) - Difference of squares: x² - 49 = x² - 7²
  8. A) (x + 2)(x - 7) - 2 × (-7) = -14 and 2 + (-7) = -5
  9. D) Both A and B - First factor GCF: 3(x² - 4), then difference of squares: 3(x + 2)(x - 2)
  10. A) (x + 7)(x - 3) - 7 × (-3) = -21 and 7 + (-3) = 4

Check Your Understanding

Answer these reflection questions to deepen your understanding.

1. Explain how the Zero Product Property connects factoring to solving equations.

Reveal Answer

The Zero Product Property states that if ab = 0, then a = 0 or b = 0. When we factor a quadratic equation like x² + 5x + 6 = 0 into (x + 2)(x + 3) = 0, we can set each factor equal to zero: x + 2 = 0 gives x = -2, and x + 3 = 0 gives x = -3. These are the solutions to the equation. Without factoring, finding these solutions would be much harder.

2. Why can't you factor x² + 9 using the difference of squares pattern?

Reveal Answer

The difference of squares pattern only works when you have a SUBTRACTION (difference) between two perfect squares: a² - b². The expression x² + 9 has an ADDITION, making it a sum of squares. The sum of squares (a² + b²) cannot be factored using real numbers. So x² + 9 is prime (cannot be factored) over the real numbers.

3. How do you determine the signs inside the factors when factoring x² + bx + c?

Reveal Answer

Look at the signs of b and c: (1) If c is positive and b is positive, both factors have positive signs: (x + )(x + ). (2) If c is positive and b is negative, both factors have negative signs: (x - )(x - ). (3) If c is negative, one factor is positive and one is negative: (x + )(x - ), where the larger number takes the same sign as b.

4. Why is it important to always check your factored answer by multiplying?

Reveal Answer

Checking by multiplication (using FOIL or distribution) verifies that your factored form is correct. It's easy to make sign errors or choose wrong factor pairs. By multiplying your answer, you confirm that it equals the original expression. This habit catches mistakes before they affect your final answer, especially important on tests where you can't afford careless errors.

🚀 Next Steps

  • Review any concepts that felt challenging
  • Move on to the next lesson when ready
  • Return to practice problems periodically for review