Algebraic identities

An identity

An equation is true only for some values of the variable. An identity is true for all values. The identities of this lesson turn long work into short work.

Identities for squares

1. $(x + y)^2 = x^2 + 2xy + y^2$
2. $(x - y)^2 = x^2 - 2xy + y^2$
3. $x^2 - y^2 = (x + y)(x - y)$
4. $(x + a)(x + b) = x^2 + (a + b)x + ab$
5. $(x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx$

The picture shows why the first one is true. A square of side $x + y$ is made of a square $x^2$, a square $y^2$, and two rectangles $xy$.

Change $a$ and $b$. The two rectangles always have the same area, $ab$. That is where the $2ab$ in $(a + b)^2$ comes from.

Identities for cubes

6. $(x + y)^3 = x^3 + y^3 + 3xy(x + y)$
7. $(x - y)^3 = x^3 - y^3 - 3xy(x - y)$
8. $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$
9. $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
10. $x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$

Using the identities

Quick calculations.

• $104 \times 96 = (100 + 4)(100 - 4) = 100^2 - 4^2 = 10000 - 16 = 9984$.
• $105^2 = (100 + 5)^2 = 10000 + 1000 + 25 = 11025$.
• $99^3 = (100 - 1)^3 = 1000000 - 30000 + 300 - 1 = 970299$.

Expanding. $(2x + 3y)^2 = 4x^2 + 12xy + 9y^2$.

Factorising. $8x^3 + 27y^3 = (2x)^3 + (3y)^3 = (2x + 3y)(4x^2 - 6xy + 9y^2)$.

A special case. If $x + y + z = 0$, then the second bracket in identity $10$ does not matter. The product is $0$, so $x^3 + y^3 + z^3 = 3xyz$.

Example: $x = -12$, $y = 7$, $z = 5$. Their sum is $0$. So $(-12)^3 + 7^3 + 5^3 = 3 \times (-12) \times 7 \times 5 = -1260$.

Practise with new numbers

Try it in Python

This program checks four identities with random numbers, and one false statement.

Remember these

• $(x + y)^2 = x^2 + 2xy + y^2$, $(x - y)^2 = x^2 - 2xy + y^2$, and $x^2 - y^2 = (x + y)(x - y)$.
• $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ and $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$.
• If $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Use an identity to find $98 \times 102$.

2. Expand $(x - 3)^2$.

3. Factorise $x^3 - 8$.

4. Expand $(2x + 1)(2x - 1)$.