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Glacius
FoundationsConcept reference

Distributing products

The distributive property states that a scalar multiplied by a sum equals the sum of its products with each term.

1 min read1 figureSources & further reading
On this page 8 sections
  1. Overview
  2. A product outside parentheses multiplies every term inside
  3. Check your understanding
  4. Check your understanding
  5. Key takeaway
  6. Further questions
  7. Sources & further reading
  8. Concept connections

01A product outside parentheses multiplies every term inside#

A product outside parentheses multiplies every term inside. Three groups of (x+2)(x+2) contain three copies of xx and three copies of 22.

Three identical groups each contain x plus 2, giving three copies of x and six units in total.Three identical groups each contain x plus 2, giving three copies of x and six units in total.
Figure 1Three identical groups each contain x plus 2, giving three copies of x and six units in total.
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Multiply 33 by xx and by 22 to get 3x+63x+6. Expanding removes the parentheses and keeps the value the same for every xx.

3(x+2)=3x+63(x+2)=3x+6

Signs belong to their terms. In −2(x−3)-2(x-3), multiply −2-2 by xx and by −3-3. The second product is positive.

−2(x−3)=−2x+6-2(x-3)=-2x+6
Check your reasoning

Which expression expands 4(x+3)4(x+3)?

  1. A4x+34x+3
  2. B4x+124x+12
  3. Cx+12x+12
Show answer and explanation
4x+124x+12

Multiply both terms by 4: 4x+4×3=4x+124x+4\times3=4x+12.

02Check your understanding#

Check your reasoning

After expanding 3(x+5)3(x+5), what constant is added to 3x3x?

Show answer and explanation
15

The outside factor also multiplies the constant: 3×5=153\times5=15.

03Check your understanding#

Check your reasoning

Each of 6 boxes holds xx pens and 2 pencils. Total objects?

  1. A6x+26x+2
  2. Bx+12x+12
  3. C6x+126x+12
Show answer and explanation
6x+126x+12

Multiply both counts by 66: 6x+126x+12.

Key takeaway

The outside factor multiplies every term inside the parentheses.

  • Expand a scalar product over a sum

Further questions

How can I check an expansion?
Choose an input such as x = 2 and evaluate both forms. A mismatch proves an error, though one matching input alone does not prove they agree for every x.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Distributing products.” Math behind ML. /learn/f-distribute