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Simplifying an expression and factoring out a common factor

Level 3 · Ages 11–13

Quick recall· Let's wake up what you already know before we start.

Smart computing with distributivity: what is 4 × 13, seen as 4 × (10 + 3)?

The whole lesson

Read it again any time, including while you work through the exercises.

Picture your bag after a treasure hunt: 5 gold coins, 3 rubies, 2 more gold coins, 1 ruby. To sort things out, you group by family: 7 gold coins and 4 rubies. Simplifying an algebraic expression is exactly that sorting: in 5x + 3 + 2x + 1, the x terms go together (5x + 2x = 7x) and the plain numbers go together (3 + 1 = 4). Result: 7x + 4.

The golden rule: you may only group terms of the same family. 5x + 2x = 7x is 5 bundles of x plus 2 bundles of x. But 3x + 5 cannot be simplified: 3 bundles of x and 5 units are different objects, like apples and pears. The expression stays 3x + 5, and that is perfectly fine.

The second tool of this lesson: factoring out a common factor. It is distributivity read backwards. You know that 4 × (3x + 2) = 12x + 8. Factoring means walking the path in reverse: in 12x + 8, spotting that 12x = 4 × 3x and 8 = 4 × 2, then pulling the 4 out front: 12x + 8 = 4(3x + 2).

Why factor? Because a product form is often far more convenient: for quick mental math, for simplifying, and soon for solving equations. A precious reflex: after factoring, expand again in your head to check you land back on the starting expression. That is your safety net.

Mind the signs: in 6x + 2 - x + 5, the sign in front of a term belongs to it. We group 6x - x = 5x and 2 + 5 = 7. Keep each sign glued to its term and all will go well.

The worked example, step by step

Simplify the expression 5x + 3 + 2x - 1.

  1. 5x + 2x = 7x

    I group the x family terms: 5 bundles of x plus 2 bundles of x make 7 bundles of x.

  2. 3 - 1 = 2

    I group the plain numbers, keeping each sign glued to its term: + 3 and - 1 give + 2.

  3. 5x + 3 + 2x - 1 = 7x + 2

    I put the two families together. We cannot go further: 7x and 2 are not from the same family.

Result: 5x + 3 + 2x - 1 = 7x + 2

Factor the expression 12x + 8.

  1. 12 = 4 × 3 et 8 = 4 × 2

    I look for the largest number that divides both 12 and 8: it is 4, my common factor.

  2. 12x + 8 = 4 × 3x + 4 × 2

    I rewrite each term to make the factor 4 appear.

  3. 4 × 3x + 4 × 2 = 4(3x + 2)

    I pull the 4 out front of the brackets: distributivity in reverse.

  4. vérif : 4(3x + 2) = 12x + 8 ✓

    I expand again in my head to check: I land back on the starting expression.

Result: 12x + 8 = 4(3x + 2)