Using Distributive Property to Factor Numerical Expressions

A step-by-step guide to using the distributive property to factor numerical expressions

To factor numerical expressions, you have to find the greatest common factor (GCF) of the monomial terms.

To rewrite the polynomial as the product of the greatest common factor (GCF) and the additional parts of the polynomial, you have to use the distributive property.

“Distribute” means to divide something or give a part of something.

The rule of the distributive property is that multiplying the sum of two or more addends via a number equals multiplying each of the addends individually by the number and then adding the products together. Both of them give the same answer.
\(a(b+c)=a×b+a×c\)

Using Distributive Property to Factor Numerical Expressions – Example 1

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Use distributive property, factor the expression, \(40+25\).
Solution:
Step 1: Find the greatest common factor of 40 and 25. It is 5.
Step 2: Divide each number by \(5. 40÷5=8, 25÷5=5.\)
Step 3: Use the distributive property to write an equivalent expression.
\(40+25=5×8+5×5=5(8+5)\)

Using Distributive Property to Factor Numerical Expressions – Example 2

Use distributive property, factor the expression, \(56+21\).
Solution:
Step 1: Find the greatest common factor of 56 and 21. It is 7.
Step 2: Divide each number by \(7. 56÷7=8, 21÷7=3\).
Step 3: Use the distributive property to write an equivalent expression.
\(56+21=7×8+7×3=7(8+3)\)

Using the Distributive Property to Factor Numerical Expressions

Factoring is the process of breaking down an expression into its component parts—the numbers or quantities you’d multiply together to get the original expression. The distributive property is your fundamental tool for factoring numerical expressions. By recognizing common factors and “factoring them out,” you transform addition into multiplication, revealing the structure within seemingly complex expressions.

Understanding the Distributive Property

The distributive property states: \(a(b + c) = ab + ac\). Reading right-to-left for factoring: \(ab + ac = a(b + c)\). The value \(a\) is the common factor that “distributes” across both terms. When factoring, you’re reversing this process—recognizing \(a\) as the common element and pulling it out front.

Finding the Greatest Common Factor (GCF)

To factor an expression using the distributive property, first identify the greatest common factor (GCF) of all terms. The GCF is the largest number that divides evenly into each term.

Example: In the expression \(12 + 18\), what is the GCF?

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common factors: 1, 2, 3, 6
• Greatest common factor: 6

The Factoring Process: Step-by-Step

Step 1: Identify All Terms

List every term in your expression. In \(15 + 25 + 10\), the three terms are 15, 25, and 10.

Step 2: Find the GCF

Determine what number divides evenly into all terms. For 15, 25, and 10:

• 15 = 3 × 5
• 25 = 5 × 5
• 10 = 2 × 5

Step 3: Divide Each Term by the GCF

Divide every term by the GCF:

• 15 ÷ 5 = 3
• 25 ÷ 5 = 5
• 10 ÷ 5 = 2

Step 4: Write the Factored Form

The factored form is GCF × (sum of quotients): \(15 + 25 + 10 = 5(3 + 5 + 2)\)

Step 5: Verify by Distributing

Check your work: \(5(3 + 5 + 2) = 5(3) + 5(5) + 5(2) = 15 + 25 + 10\) ✓

Worked Examples: Factoring Numerical Expressions

Example 1: Simple GCF Factoring

Problem: Factor \(12 + 18\)

Solution:

• Find GCF of 12 and 18: Both are divisible by 6 (and also by 1, 2, 3, but 6 is greatest)
• Divide each term: \(12 ÷ 6 = 2\) and \(18 ÷ 6 = 3\)
• Factored form: \(6(2 + 3)\)
• Check: \(6(2 + 3) = 12 + 18\) ✓

Example 2: Larger Numbers with Multiple Terms

Problem: Factor \(28 + 42 + 14\)

Solution:

• Prime factorizations:
  • 28 = 2² × 7
  • 42 = 2 × 3 × 7
  • 14 = 2 × 7
• Common factors in all three: 2 and 7, so GCF = 2 × 7 = 14
• Divide each: \(28 ÷ 14 = 2\), \(42 ÷ 14 = 3\), \(14 ÷ 14 = 1\)
• Factored form: \(14(2 + 3 + 1)\)
• Check: \(14(2 + 3 + 1) = 28 + 42 + 14\) ✓

Example 3: Factoring with Subtraction

Problem: Factor \(48 – 32\)

Solution:

• GCF of 48 and 32:
  • 48 = 2⁴ × 3
  • 32 = 2⁵
  Common: 2⁴ = 16
• Divide each: \(48 ÷ 16 = 3\) and \(32 ÷ 16 = 2\)
• Factored form: \(16(3 – 2)\)
• Check: \(16(3 – 2) = 16(1) = 16 = 48 – 32\) ✓

Example 4: Multiple-Term Expression

Problem: Factor \(36 + 54 + 72 + 18\)

Solution:

• Find GCF:
  • 36 = 2² × 3²
  • 54 = 2 × 3³
  • 72 = 2³ × 3²
  • 18 = 2 × 3²
  Common: 2 × 3² = 18
• Divide each: \(36 ÷ 18 = 2\), \(54 ÷ 18 = 3\), \(72 ÷ 18 = 4\), \(18 ÷ 18 = 1\)
• Factored form: \(18(2 + 3 + 4 + 1)\)
• Check: \(18(2 + 3 + 4 + 1) = 18(10) = 180 = 36 + 54 + 72 + 18\) ✓

Why Factoring Matters

Simplification

Factored form often reveals structure. Instead of thinking of \(24 + 40\) as two separate numbers, seeing it as \(8(3 + 5)\) shows the underlying relationship clearly.

Mental Math

Factoring simplifies calculations. \(8(3 + 5) = 8(8) = 64\) is often easier to compute than \(24 + 40\).

Algebra Foundation

Factoring numerical expressions is the foundation for factoring algebraic expressions like \(6x + 9\), which factors to \(3(2x + 3)\) using the exact same process.

Common Factoring Patterns

Pattern 1: Factor of 2 or 5 Expressions like \(14 + 26 + 30\) often have 2 as a factor. Check: \(14 + 26 + 30 = 2(7 + 13 + 15)\).

Pattern 2: Multiples of 10 Numbers ending in 0 suggest trying 10 as a factor. For \(30 + 50 + 20 = 10(3 + 5 + 2)\).

Pattern 3: Numbers in a sequence Expressions like \(10 + 20 + 30\) = \(10(1 + 2 + 3)\) show the power of recognizing repeated structure.

Practice Problems

1. Factor \(16 + 24\)
2. Factor \(35 + 49 + 21\)
3. Factor \(60 – 45\)
4. Factor \(27 + 36 + 18 + 45\)
5. Factor \(100 + 150 + 75\)

Using the Distributive Property to Factor Numerical Expressions

Factoring is the process of breaking down an expression into its component parts—the numbers or quantities you’d multiply together to get the original expression. The distributive property is your fundamental tool for factoring numerical expressions. By recognizing common factors and “factoring them out,” you transform addition into multiplication, revealing the structure within seemingly complex expressions.

Understanding the Distributive Property

The distributive property states: \(a(b + c) = ab + ac\). Reading right-to-left for factoring: \(ab + ac = a(b + c)\). The value \(a\) is the common factor that “distributes” across both terms. When factoring, you’re reversing this process—recognizing \(a\) as the common element and pulling it out front.

Finding the Greatest Common Factor (GCF)

To factor an expression using the distributive property, first identify the greatest common factor (GCF) of all terms. The GCF is the largest number that divides evenly into each term.

Example: In the expression \(12 + 18\), what is the GCF?

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common factors: 1, 2, 3, 6
• Greatest common factor: 6

The Factoring Process: Step-by-Step

Step 1: Identify All Terms

List every term in your expression. In \(15 + 25 + 10\), the three terms are 15, 25, and 10.

Step 2: Find the GCF

Determine what number divides evenly into all terms. For 15, 25, and 10:

• 15 = 3 × 5
• 25 = 5 × 5
• 10 = 2 × 5

Step 3: Divide Each Term by the GCF

Divide every term by the GCF:

• 15 ÷ 5 = 3
• 25 ÷ 5 = 5
• 10 ÷ 5 = 2

Step 4: Write the Factored Form

The factored form is GCF × (sum of quotients): \(15 + 25 + 10 = 5(3 + 5 + 2)\)

Step 5: Verify by Distributing

Check your work: \(5(3 + 5 + 2) = 5(3) + 5(5) + 5(2) = 15 + 25 + 10\) ✓

Worked Examples: Factoring Numerical Expressions

Example 1: Simple GCF Factoring

Problem: Factor \(12 + 18\)

Solution:

• Find GCF of 12 and 18: Both are divisible by 6 (and also by 1, 2, 3, but 6 is greatest)
• Divide each term: \(12 ÷ 6 = 2\) and \(18 ÷ 6 = 3\)
• Factored form: \(6(2 + 3)\)
• Check: \(6(2 + 3) = 12 + 18\) ✓

Example 2: Larger Numbers with Multiple Terms

Problem: Factor \(28 + 42 + 14\)

Solution:

• Prime factorizations:
  • 28 = 2² × 7
  • 42 = 2 × 3 × 7
  • 14 = 2 × 7
• Common factors in all three: 2 and 7, so GCF = 2 × 7 = 14
• Divide each: \(28 ÷ 14 = 2\), \(42 ÷ 14 = 3\), \(14 ÷ 14 = 1\)
• Factored form: \(14(2 + 3 + 1)\)
• Check: \(14(2 + 3 + 1) = 28 + 42 + 14\) ✓

Example 3: Factoring with Subtraction

Problem: Factor \(48 – 32\)

Solution:

• GCF of 48 and 32:
  • 48 = 2⁴ × 3
  • 32 = 2⁵
  Common: 2⁴ = 16
• Divide each: \(48 ÷ 16 = 3\) and \(32 ÷ 16 = 2\)
• Factored form: \(16(3 – 2)\)
• Check: \(16(3 – 2) = 16(1) = 16 = 48 – 32\) ✓

Example 4: Multiple-Term Expression

Problem: Factor \(36 + 54 + 72 + 18\)

Solution:

• Find GCF:
  • 36 = 2² × 3²
  • 54 = 2 × 3³
  • 72 = 2³ × 3²
  • 18 = 2 × 3²
  Common: 2 × 3² = 18
• Divide each: \(36 ÷ 18 = 2\), \(54 ÷ 18 = 3\), \(72 ÷ 18 = 4\), \(18 ÷ 18 = 1\)
• Factored form: \(18(2 + 3 + 4 + 1)\)
• Check: \(18(2 + 3 + 4 + 1) = 18(10) = 180 = 36 + 54 + 72 + 18\) ✓

Why Factoring Matters

Simplification

Factored form often reveals structure. Instead of thinking of \(24 + 40\) as two separate numbers, seeing it as \(8(3 + 5)\) shows the underlying relationship clearly.

Mental Math

Factoring simplifies calculations. \(8(3 + 5) = 8(8) = 64\) is often easier to compute than \(24 + 40\).

Algebra Foundation

Factoring numerical expressions is the foundation for factoring algebraic expressions like \(6x + 9\), which factors to \(3(2x + 3)\) using the exact same process.

Common Factoring Patterns

Pattern 1: Factor of 2 or 5 Expressions like \(14 + 26 + 30\) often have 2 as a factor. Check: \(14 + 26 + 30 = 2(7 + 13 + 15)\).

Pattern 2: Multiples of 10 Numbers ending in 0 suggest trying 10 as a factor. For \(30 + 50 + 20 = 10(3 + 5 + 2)\).

Pattern 3: Numbers in a sequence Expressions like \(10 + 20 + 30\) = \(10(1 + 2 + 3)\) show the power of recognizing repeated structure.

Practice Problems

1. Factor \(16 + 24\)
2. Factor \(35 + 49 + 21\)
3. Factor \(60 – 45\)
4. Factor \(27 + 36 + 18 + 45\)
5. Factor \(100 + 150 + 75\)