Expanding brackets

Like terms, the distributive law, expanding binomial products and the three special expansions: perfect squares and the difference of squares.

Like terms

have exactly the same pronumerals with the same powers. You can add and subtract like terms, but not unlike ones: $3x^2 + 5x^2 = 8x^2$, but $3x^2 + 5x$ cannot be simplified.

The distributive law

Every expansion comes from one law: multiply each term inside the bracket by the term outside.

Distributive law

$$a(b + c) = ab + ac \qquad (a + b)(c + d) = ac + ad + bc + bd$$

Example 1●

Expand and simplify $3(2x - 5) - 2(x - 4)$.

$= 6x - 15 - 2x + 8$

Multiply each bracket out. The −2 multiplies both x and −4.

$= 4x - 7$

Collect the x terms and the constants.

Example 2●●

Expand $(2x + 3)(x - 4)$.

$= 2x^2 - 8x + 3x - 12$

Each term in the first bracket multiplies each term in the second: four products.

$= 2x^2 - 5x - 12$

The three special expansions

Three products appear so often that you should recognise them instantly, both forwards (expanding) and backwards (factoring).

Perfect squares

$$(A + B)^2 = A^2 + 2AB + B^2 \qquad (A - B)^2 = A^2 - 2AB + B^2$$

Difference of squares

$$(A + B)(A - B) = A^2 - B^2$$

Example 3●●

Expand $(3a - 2b)^2$ and $(5x + 1)(5x - 1)$.

$(3a - 2b)^2 = 9a^2 - 12ab + 4b^2$

A = 3a, B = 2b. Square the first, double the product, square the last.

$(5x + 1)(5x - 1) = 25x^2 - 1$

Sum times difference of the same two terms: the middle terms cancel.

Check your expansion with graphs

Orange: the product of brackets. Dashed blue: your expanded answer. If your expansion is right, the two graphs are identical for every x.

Try this

Set $p$ and $q$, expand $(x + p)(x + q)$ in your head, then set "your b" and "your c" to match. The graphs tell you if you are right.

Flashcards

(A + B)²

$A^2 + 2AB + B^2$

(A − B)²

$A^2 - 2AB + B^2$

(A + B)(A − B)

$A^2 - B^2$

Like terms

Same pronumerals with the same powers.

Practice

$(x - 5)^2 =$
$(2x + 7)(2x - 7) =$
Expand and simplify $(x + 2)(3x - 1) - 3x^2$. What is the coefficient of $x$?
$-3(2x - 5) =$
Simplify $4a^2b - 7a^2b + a^2b$. Enter the coefficient of $a^2b$.
$(3x - 1)^2 =$
Expand $(2x - 3)(x + 5)$. What is the coefficient of $x$?
Which pair are like terms?