Factor Linear Expressions

Grade 7 · algebra · 93 practice problems · read aloud

🔊 Listen to this explanation

Factor Linear Expressions

🎯 What is Factoring?

Factoring is like "un-multiplying." You take an expression like 4x + 12 and rewrite it as 4(x + 3). You're finding the greatest common factor (GCF) that divides evenly into all terms.

Why it's useful: It simplifies expressions, helps solve equations, and is a key skill for all of algebra!

📝 Step-by-Step Guide

  1. Find the GCF: Look at the coefficients (numbers) and variables in all terms. Find the largest number and any common variables that divide into everything.
  2. Divide each term by the GCF: Take your original terms and divide them by the GCF you found.
  3. Write the factored form: Place the GCF outside parentheses, and the divided results inside. Check your work by distributing!

🔍 Worked Examples

Example 1: Factor 6x + 9

Step 1: GCF of 6 and 9 is 3. No common variable.

Step 2: 6x ÷ 3 = 2x, and 9 ÷ 3 = 3.

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

✅ Check: 3 × 2x = 6x, 3 × 3 = 9. Correct!

Example 2: Factor 15y - 25

Step 1: GCF of 15 and 25 is 5.

Step 2: 15y ÷ 5 = 3y, 25 ÷ 5 = 5.

Step 3: Answer: 5(3y - 5)

Example 3: Factor 8m + 12m²

Step 1: GCF of 8 and 12 is 4. Both terms have at least one 'm', so GCF is 4m.

Step 2: 8m ÷ 4m = 2, 12m² ÷ 4m = 3m.

Step 3: Answer: 4m(2 + 3m)

⚠️ Common Mistakes

Forgetting the 1: When factoring 5x + 5, it's 5(x + 1), not 5(x + 0).

Not factoring completely: For 12x + 18, the GCF is 6, not 2. Always find the greatest common factor.

Losing negative signs: In -4x - 8, the factored form is -4(x + 2). The negative sign must be included!

💡 Tips & Tricks

GCF Rainbow Method: List factor pairs of each number to visually find the largest common one.

Variable Rule: For variables, use the one with the smallest exponent that appears in all terms.

Always Check: Multiply your factored form back out. If you get the original expression, you're right!

🏋️ Practice Suggestions

  • Start simple: Practice finding the GCF of number pairs (like 24 and 36).
  • Create flashcards with expressions on one side and factored forms on the other.
  • Try these: Factor 9n + 15, 20 - 16k, 7x² + 14x. (Answers: 3(3n+5), 4(5-4k), 7x(x+2))
  • Work with a partner—one writes an expression, the other factors it!

Practice problems

6 of the 93, worked through step by step — try them before opening the answer.

1 Factor: 48x + 72 = ?

Hint: Look for the largest even number that divides evenly into both 48 and 72. Then think about what each term becomes when divided by that number.

Show the answer

Answer: 24(2x + 3)

  1. Identify the coefficients: 48 and 72.
  2. Find the GCF of 48 and 72. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 The GCF is 24.
  3. Divide each term by the GCF. 48x ÷ 24 = 2x 72 ÷ 24 = 3
  4. Write the factored form: 24(2x + 3)

The answer is 24(2x + 3).

2 Factor: 91x + 26 = ?

Hint: Look for the largest number that divides evenly into both 91 and 26. Think about the factors of each number and find their greatest common factor.

Show the answer

Answer: 13(7x + 2)

  1. Identify the coefficients: 91 and 26.
  2. Find the GCF of 91 and 26. - Factors of 91: 1, 7, 13, 91 - Factors of 26: 1, 2, 13, 26 - The GCF is 13.
  3. Divide each term by 13. - 91x ÷ 13 = 7x - 26 ÷ 13 = 2
  4. Write the factored form: 13(7x + 2)

The answer is 13(7x + 2).

3 Factor: 84x - 63 = ?

Hint: Look for the largest number that divides evenly into both the coefficient of x and the constant term. Then rewrite each term as a product of that number and something else.

Show the answer

Answer: 21(4x - 3)

  1. Identify the terms: 84x and -63.
  2. Find the GCF of 84 and 63. Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 Factors of 63: 1, 3, 7, 9, 21, 63 The greatest common factor is 21.
  3. Divide each term by the GCF. 84x ÷ 21 = 4x -63 ÷ 21 = -3
  4. Write the factored form: 21(4x - 3).

The answer is 21(4x - 3).

4 Factor: 84x + 126 = ?

Hint: Look for the largest number that divides evenly into both coefficients. Think about the multiplication table for each coefficient and find their common factor.

Show the answer

Answer: 42(2x + 3)

  1. Identify the coefficients: 84 and 126.
  2. Find the GCF of 84 and 126. - Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84. - Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126. - The greatest common factor is 42.
  3. Divide each term by the GCF. - 84x ÷ 42 = 2x - 126 ÷ 42 = 3
  4. Write the factored expression: 42(2x + 3).

The answer is 42(2x + 3).

5 Factor: 144x - 96 = ?

Hint: Find the largest number that divides evenly into both 144 and 96. Think about the factors of each number and identify their greatest common factor.

Show the answer

Answer: 48(3x - 2)

  1. Identify the terms: 144x and -96.
  2. Find the GCF of 144 and 96. Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 Factors of 96: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 The greatest common factor is 48.
  3. Divide each term by the GCF. 144x ÷ 48 = 3x -96 ÷ 48 = -2
  4. Write the factored form: 48(3x - 2).

The answer is 48(3x - 2).

6 Factor: 112x + 77 = ?

Hint: Look for the largest number that divides evenly into both the coefficient of x and the constant term. Think about the multiplication table of the smaller number to find a common factor.

Show the answer

Answer: 7(16x + 11)

  1. Identify the terms: 112x and 77.
  2. Find the GCF of 112 and 77. Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112 Factors of 77: 1, 7, 11, 77 The greatest common factor is 7.
  3. Divide each term by the GCF. 112x ÷ 7 = 16x 77 ÷ 7 = 11
  4. Write the factored form: 7(16x + 11).

The answer is 7(16x + 11).

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