Algebraic identities — (a+b)², (a−b)², (a+b)(a−b), (a+b)³, (a−b)³, (a³+b³), (a³−b³): Foundations

This page assumes you have seen nothing. We build every symbol the parent note Algebraic identities leans on, in an order where each block sits on the one before it.

1. A letter that stands for a number (a variable)

The picture: imagine an empty box with a label on it. The label $a$ does not tell you what is inside; it only promises "one fixed number lives here." Whatever number you drop in, you must use the same number everywhere that label appears.

Why the topic needs it: an identity like $(a + b)^{2} = a^{2} + 2 ab + b^{2}$ is a promise about all numbers at once. Without a letter to stand for "any number", we would have to write a separate line for $1$ , for $2$ , for $3.7$ , forever. One letter captures infinitely many cases in a single statement.

Figure — Algebraic identities — (a+b)², (a−b)², (a+b)(a−b), (a+b)³, (a−b)³, (a³+b³), (a³−b³)

2. Putting a number in front of a letter — multiplication

The picture: $2 a$ is two copies of the box $a$ laid end to end — a strip twice as long as one $a$ . And $ab$ is a rectangle: one side has length $a$ , the other side has length $b$ , and $ab$ is the area inside.

3. The little raised number — a power

Why the words "squared" and "cubed"? Look at the picture, not just the rule:

$a^{2}$ is the area of a square whose side is $a$ . That is literally why we say "squared".
$a^{3}$ is the volume of a cube whose side is $a$ . That is why we say "cubed".

Why the topic needs powers: the identities are named after squares and cubes — $(a + b)^{2}$ , $(a + b)^{3}$ , $a^{3} + b^{3}$ . Every one is a statement about areas or volumes of shapes built from $a$ and $b$ .

4. Plus and minus inside a bracket — a binomial

The picture: $(a + b)$ is a single line segment made of two pieces glued in a row — a piece of length $a$ followed by a piece of length $b$ . Its total length is $a + b$ .

What the minus means as a picture: $(a - b)$ is the length $a$ with a piece of length $b$ cut off the end. So $a - b$ is what remains. (This only makes visual sense when $a$ is at least as big as $b$ , but the algebra still works for any numbers — a topic we return to below.)

Why the topic needs it: every one of the seven identities starts with a binomial being squared, cubed, or multiplied by another binomial. The bracket is what tells you "do the plus/minus first, then apply the power to the whole thing."

The picture: think of a balance scale that stays perfectly level no matter what weights you place — because both pans always hold the same amount, just described two different ways (folded up as $(a + b)^{2}$ on the left, unfolded as $a^{2} + 2 ab + b^{2}$ on the right).

Why the topic needs it: the whole point of an identity is that it never fails. That is why we can use it as a shortcut — we never have to re-check it for the particular numbers in front of us.

6. The tool that makes them all work — the distributive property

Why this tool and not another? Multiplying $(a + b) (a + b)$ has no meaning by itself until we know how to spread one bracket across the other. The distributive property is the only rule that tells us that — it is the engine behind every single derivation in the parent note. Every step that says "distributive property" is just this one picture, reused.

The picture: $a (b + c)$ is a rectangle of height $a$ sitting on a base split into a $b$ part and a $c$ part. Its total area ( $a (b + c)$ ) is clearly the two smaller rectangles added: $ab + a c$ .

7. "Like terms" — what you may combine

The picture: you can add two rectangles of the same shape ( $ab + ab = 2 ab$ ) because they stack into one object. You cannot merge a square ( $a^{2}$ ) with a rectangle ( $ab$ ) — they are different shapes, so $a^{2} + ab$ stays as two separate pieces.

Why the topic needs it: the famous middle term $2 ab$ in $(a + b)^{2}$ appears because two identical $ab$ rectangles get combined. Recognising like terms is the final step of nearly every derivation.

How these foundations feed the topic

Read it top to bottom: a variable lets us talk about all numbers; multiplication and powers build squares and cubes; binomials glue two variables together; the distributive property (as FOIL) multiplies binomials out; combining like terms tidies the result — and only then do the seven identities appear.

Equipment checklist

Test yourself — answer out loud, then reveal.

What does the letter

a

mean in an identity?

A placeholder box holding one fixed number; the same number everywhere

a

appears, but any number you like.

What does

2 a

mean, and what picture goes with it?

2

times

a

; two copies of the

a

strip laid end to end.

What does

ab

mean as a picture?

The area of a rectangle with sides

a

and

b

What does the raised

2

a^{2}

count?

The number of copies of

a

multiplied together —

a \times a

, the area of a square of side

a

Why is

a^{2}

read "

a

squared"?

Because it is the area of a square whose side is

a

What is a binomial?

Two terms joined by

+

-

inside brackets, treated as one object, e.g.

(a - b)

What is the difference between

(a + b)^{2}

and

a + b^{2}

The bracket squares the whole sum; without it only

b

is squared. They give different answers.

How does an identity differ from an equation?

An equation is true for special values only; an identity is true for every value — a permanent promise.

State the distributive property.

a (b + c) = ab + a c

: multiply the outside by every term inside and add.

When may two terms be combined into one?

Only when they are like terms — same letters, same powers, e.g.

ab + ab = 2 ab