Distribute a factor and reverse the operation

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Distribute a factor and reverse the operation

A factor outside a bracket multiplies every term inside. Thus 3(x+2)=3x+6. A negative multiplier changes each sign: −2(x−4)=−2x+8. Expanding means replacing the product with its equivalent sum.

Factorising reverses distribution. Find a common factor of every term and divide each term by it inside the bracket. For 6x+9, factorising 3 gives 3(2x+3). Expanding the answer is a direct check that no term or sign was lost.

An area model can make distribution visible for positive lengths, but the algebraic identity is more general. A rectangle with side x and total width x+3 has areas x² and 3x. It illustrates x(x+3)=x²+3x when x is positive; negative values are handled algebraically, not as physical lengths.

Worked example: Expand −3(2x−5), then factorise 8x+12.

Multiply−3 by 2x and by−5.

For 8x+12, take out the common factor 4.

Answer: −6x+15;4(2x+3).

Check your understanding: Does 2(x+4) equal 2x+4?

Answer: No; it equals 2x+8.

Why: The outside factor multiplies both terms in the bracket.