Algebraic identities

Some products show up so often that mathematicians give them special names. Knowing these identities lets you skip the FOIL work and write the answer in one move. They also work in reverse , for factorising.

Concept

Identity 1: Square of a sum.

$(a + b)^2 = a^2 + 2ab + b^2.$

Why? $(a+b)(a+b) = a^2 + ab + ab + b^2 = a^2 + 2ab + b^2$.

Identity 2: Square of a difference.

$(a - b)^2 = a^2 - 2ab + b^2.$

Same FOIL with $b$ replaced by $-b$. Note the middle term is minus $2ab$, while the last term $(-b)^2 = b^2$ is positive.

Identity 3: Difference of squares.

$(a + b)(a - b) = a^2 - b^2.$

Why? $(a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2$. The middle terms cancel.

Identity 4: Cube of a sum.

$(a + b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3.$

Identity 5: Cube of a difference.

$(a - b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3.$

Identity 6: Sum and difference of cubes.

$a^3 + b^3 = (a+b)(a^2 - ab + b^2), \qquad a^3 - b^3 = (a-b)(a^2 + ab + b^2).$

These are the ones used most often in class VIII. We will see more later when you study quadratics in detail.

Why memorise? Because algebra problems often hide one of these identities inside them. The moment you spot it, the problem becomes a one-step computation.

Example: compute $97 \times 103$ without a calculator.

• Recognise $97 \times 103 = (100 - 3)(100 + 3)$, which is a difference of squares.
• $= 100^2 - 3^2 = 10000 - 9 = 9991$.

Example: compute $105^2$.

• $105^2 = (100 + 5)^2 = 100^2 + 2(100)(5) + 5^2 = 10000 + 1000 + 25 = 11025$.

The identity is a shortcut not just for algebra but for arithmetic.

Worked examples

Example 1. Expand $(2x + 3)^2$.

• Use $(a+b)^2 = a^2 + 2ab + b^2$ with $a = 2x, b = 3$.
• $= 4x^2 + 12x + 9$.

Example 2. Expand $(5a - 4)^2$.

• Use $(a-b)^2$.
• $= 25a^2 - 40a + 16$.

Example 3. Compute $(x + y)(x - y)$ when $x = 13$ and $y = 7$ without separate multiplication.

• $(x+y)(x-y) = x^2 - y^2 = 169 - 49 = 120$.

Example 4. Expand $(x + 2)^3$.

• $= x^3 + 3 x^2 (2) + 3 x (4) + 8 = x^3 + 6x^2 + 12x + 8$.

Try it yourself

1. Expand $(3x + 4)^2$.
2. Expand $(a - 7)^2$.
3. Expand $(2p + 5q)(2p - 5q)$.
4. Compute $102^2$ using an identity.
5. Compute $99 \times 101$ using an identity.
6. Expand $(x - 1)^3$.
7. Show that $(a + b)^2 - (a - b)^2 = 4ab$.
8. Show that $a^3 - b^3$ when $a = 5, b = 3$ equals $(a-b)(a^2 + ab + b^2)$ both computed and predicted.

Activity / Insight

Identity from a square. Draw a square of side $(a + b)$. Cut it into four pieces: a square of side $a$, two rectangles of $a \times b$, and a square of side $b$. The total area is $(a+b)^2$ and the four pieces sum to $a^2 + 2ab + b^2$. You have just seen identity 1. Try a similar geometric proof for $(a-b)^2$ , it's a bigger square minus the pieces you cut away.