Factoring Expressions

What is Factoring?

Factoring is the reverse of multiplying—writing an expression as a product.

Multiply: 3(x + 2) = 3x + 6 Factor: 3x + 6 = 3(x + 2)

Why factor? Makes solving equations easier!

Greatest Common Factor (GCF)

The GCF is the largest factor shared by all terms.

To find GCF:

1. Find GCF of coefficients
2. Find lowest power of each variable
3. Multiply them together

Example 1: Find GCF

Terms: 12x³ and 18x²

Coefficients: GCF(12, 18) = 6 Variables: Lowest power of x is x²

GCF: 6x²

Example 2: Three Terms

Expression: 15x⁴ + 10x³ − 5x²

Coefficients: GCF(15, 10, 5) = 5 Variables: Lowest power is x²

GCF: 5x²

Factoring Out the GCF

Steps:

1. Find the GCF
2. Divide each term by GCF
3. Write as: GCF(quotient)

Example 1: Factor 6x + 9

Step 1: Find GCF

• GCF(6, 9) = 3

Step 2: Divide each term by 3

• 6x ÷ 3 = 2x
• 9 ÷ 3 = 3

Step 3: Write factored form

• 3(2x + 3)

Answer: 3(2x + 3)

Check: 3(2x + 3) = 6x + 9 ✓

Example 2: Factor 4x² + 8x

GCF: 4x

Divide:

• 4x² ÷ 4x = x
• 8x ÷ 4x = 2

Answer: 4x(x + 2)

Check: 4x(x + 2) = 4x² + 8x ✓

Example 3: Three Terms

Factor: 12x³ − 18x² + 6x

GCF: 6x

Divide:

• 12x³ ÷ 6x = 2x²
• 18x² ÷ 6x = 3x
• 6x ÷ 6x = 1

Answer: 6x(2x² − 3x + 1)

Example 4: Negative GCF

Factor: −3x − 6

GCF: −3

Divide:

• −3x ÷ (−3) = x
• −6 ÷ (−3) = 2

Answer: −3(x + 2)

Note: Factoring out negative makes remaining terms positive!

When GCF is a Variable

Example: Factor x³ + x²

GCF: x²

Divide:

• x³ ÷ x² = x
• x² ÷ x² = 1

Answer: x²(x + 1)

Important: Don't forget the 1!

Factoring by Grouping

For four terms: Group in pairs, factor each pair

Example: Factor x³ + 3x² + 2x + 6

Step 1: Group

• (x³ + 3x²) + (2x + 6)

Step 2: Factor each group

• x²(x + 3) + 2(x + 3)

Step 3: Factor out common binomial

• (x + 3)(x² + 2)

Answer: (x + 3)(x² + 2)

Difference of Squares

Pattern: a² − b² = (a + b)(a − b)

Recognize: Two perfect squares separated by minus

Example 1: Factor x² − 9

Identify: x² and 9 are perfect squares

• x² = (x)²
• 9 = 3²

Apply pattern:

• x² − 9 = (x + 3)(x − 3)

Answer: (x + 3)(x − 3)

Check: (x + 3)(x − 3) = x² − 9 ✓

Example 2: Factor 4x² − 25

Identify squares:

• 4x² = (2x)²
• 25 = 5²

Factor:

• (2x + 5)(2x − 5)

Answer: (2x + 5)(2x − 5)

Example 3: Not Factorable

x² + 9

Note: Sum of squares (plus sign) Cannot factor using real numbers!

Only difference of squares factors easily

Perfect Square Trinomials

Patterns:

• a² + 2ab + b² = (a + b)²
• a² − 2ab + b² = (a − b)²

Example 1: Factor x² + 6x + 9

Recognize pattern:

• First term: x²
• Last term: 9 = 3²
• Middle: 6x = 2(x)(3) ✓

Factor: (x + 3)²

Check: (x + 3)² = x² + 6x + 9 ✓

Example 2: Factor x² − 10x + 25

Pattern check:

• x² and 25 = 5²
• Middle: −10x = −2(x)(5) ✓

Factor: (x − 5)²

Checking Your Factoring

Always multiply back to verify!

Example: Check Factoring

Given: 2x(3x − 4)

Multiply:

• 2x × 3x = 6x²
• 2x × (−4) = −8x
• Result: 6x² − 8x ✓

Original was 6x² − 8x, so factoring is correct!

Prime Polynomials

Prime polynomial: Cannot be factored (like prime numbers)

Example: x² + 2x + 3

• No two numbers multiply to 3 and add to 2
• Prime (cannot factor)

Common Mistakes

❌ Forgetting the 1:

• x² + x should be x(x + 1), not x(x)

❌ Wrong GCF:

• 6x + 9: GCF is 3, not 6

❌ Not factoring completely:

• 2x² + 4x = 2(x² + 2x) is not fully factored
• ✓ 2x(x + 2)

❌ Confusing sum and difference:

• x² + 4 cannot factor (sum of squares)
• x² − 4 = (x + 2)(x − 2) (difference)

Real-World Applications

Simplifying expressions:

• Before solving equations
• Reducing fractions

Finding dimensions:

• Area = x² + 5x = x(x + 5)
• Dimensions are x and (x + 5)

Solving equations:

• x² − 5x = 0
• x(x − 5) = 0
• Solutions: x = 0 or x = 5

Practice