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Expanding and factoring

Level 4 · Ages 13–15

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

Mentally compute 7 × 103 by splitting 103 into 100 + 3.

The whole lesson

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

Here is a mental math trick you may already use: to compute 7 × 103, you split 103 into 100 + 3, then compute 7 × 100 + 7 × 3 = 700 + 21 = 721. You just used distributivity: the 7 is distributed over each part of the sum. In algebra it works the same way: k(a + b) = ka + kb.

Expanding means turning a product into a sum: 3(x + 5) becomes 3x + 15. The 3 multiplies every term inside the brackets, without skipping any. When two brackets multiply each other, every term of the first multiplies every term of the second: (a + b)(c + d) = ac + ad + bc + bd. Four products, not two!

Factoring is the reverse journey: turning a sum into a product. In 3x + 15, the factor 3 appears in both terms, so 3x + 15 = 3(x + 5). Expanding and factoring are two directions on the same road.

Three products come up so often that they have a name: special products. (a + b)² = a² + 2ab + b², (a - b)² = a² - 2ab + b², and (a - b)(a + b) = a² - b². The classic trap: believing that (a + b)² = a² + b². That is wrong, the double product 2ab is missing! Check with a = 3 and b = 1: (3 + 1)² = 16, while 3² + 1² = 10.

These identities are also mental math superpowers: 99² = (100 - 1)² = 10000 - 200 + 1 = 9801, no written calculation needed. Your turn!

The worked example, step by step

Expand and simplify (x + 3)(x + 2).

  1. x × x = x² et x × 2 = 2x

    The first term x multiplies each term of the second bracket.

  2. 3 × x = 3x et 3 × 2 = 6

    Then the 3 multiplies each term in turn. Four products in total, none forgotten!

  3. x² + 2x + 3x + 6 = x² + 5x + 6

    I gather everything and simplify: 2x + 3x = 5x.

Result: (x + 3)(x + 2) = x² + 5x + 6

Expand (x - 5)² using the special product.

  1. (a - b)² = a² - 2ab + b²

    I recognise the form (a - b)² with a = x and b = 5.

  2. a² = x², 2ab = 2 × x × 5 = 10x, b² = 25

    I compute the three pieces: the square of x, the double product, the square of 5.

  3. (x - 5)² = x² - 10x + 25

    I assemble: minus the double product, plus the square of b. The 25 is positive because a square is always positive.

Result: (x - 5)² = x² - 10x + 25