The distributive property
Expand or factor an expression while preserving its value.
Before you begin
- Multiplication and addition
- Reading brackets
Multiply every part of a group
The distributive property says that multiplying a sum is the same as multiplying each addend and then adding the products. Three groups of x + 4 contain three lots of x and three lots of four, so 3(x + 4) = 3x + 12. A rectangle split into two widths can illustrate the same relationship: the area of the whole equals the sum of the two smaller areas.
Reverse the process to factor
Factoring collects a common multiplier into a bracket. For 8x + 12, both coefficients have a factor of 4, so write 4(2x + 3). Dividing each original term by the outside factor tells you what belongs inside. Check by expanding again. The bracket must reproduce every term, including a subtraction term when one is present.
Worked example
Expand 5(x + 3), then factor 10y + 15 using its greatest common factor.
Show the worked solution
- Multiply both terms in the first bracket: 5x + 5 × 3 = 5x + 15.
- The greatest common factor of 10 and 15 is 5.
- Divide each term by 5: 10y + 15 = 5(2y + 3). Expanding checks the result.
5x + 15; 5(2y + 3).
Try it yourself
Which expression equals 4(2 + x)?
Common mistakes
- An outside multiplier applies to every term inside the bracket.
- Factoring is not complete until expansion reproduces the original expression.
What can you explain now?
Rewrite 18a + 24 as a product using the greatest common factor.
Compare with the explanation
6(3a + 4).
The coefficient GCF is 6. Divide each term by 6, then check that distributing gives 18a + 24.
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Common Core mathematics standards ↗