How many special products are there?

School mathematics uses seven: the square of a sum, the square of a difference, the difference of squares, the cube of a sum, the cube of a difference, the difference of cubes and the sum of cubes. The first three are the core set and cover most problems; the remaining four belong to the extended syllabus.

Why is (a + b)² not a² + b²?

Because multiplying two brackets multiplies every term by every term: (a + b)(a + b) = a² + ab + ba + b². The two middle terms are the same and add up to 2ab. The shortest numerical proof: (3 + 4)² = 49, while 3² + 4² = 25. The missing 24 is exactly 2 · 3 · 4.

Is there a formula for a sum of squares a² + b²?

There is no factorization of a² + b² with real coefficients — a sum of squares is irreducible. Only the difference factors: a² − b² = (a − b)(a + b). Confusing the two is one of the most common slips in school algebra.

How do I recognize a perfect square trinomial?

Check three things: the outer terms are squares (a² and b²), they share a sign, and the middle term equals twice the product 2ab. For instance 9x² + 30x + 25 has outer terms (3x)² and 5², and 2 · 3x · 5 = 30x — it matches, so the expression is (3x + 5)².

What are special products actually for?

For two things at once. Read left to right they shorten multiplying brackets to one line. Read right to left they factor an expression, and that is the step without which you cannot cancel an algebraic fraction, state a domain or solve a polynomial inequality.

Intermediate level

Special products

Seven identities that turn multiplying out brackets into a single line — and the same seven read right to left, which is factoring. The square of a sum and of a difference, the difference of squares, the cubes and the sum and difference of cubes, proved on a drawing and used for mental arithmetic.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Algebraic expressionsAn algebraic expression is a piece of notation where letters stand beside numbers. Learn the variable, the term and the coefficient, collect like terms, multiply brackets out — two binomials included — factor out the common term and evaluate an expression.
PowersA power is shorthand for multiplying the same factor by itself. Learn the base and the exponent, the laws of exponents, powers of a product and of a quotient, zero and negative exponents, the monotonicity of exponentiation, and scientific notation.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Mental arithmetic at the till
The product 98 · 102 is (100 − 2)(100 + 2), so 10,000 − 4 = 9,996. In the same way 47 · 53 = 50² − 3² = 2,500 − 9 = 2,491. Any product of two numbers sitting symmetrically around a round hundred or fifty collapses into a single subtraction.
A plot of land and a wider garden
A square plot has a 40 m side. Widening it by 3 m on every side adds (40 + 3)² − 40² = 6 · 40 + 9 = 249 m², not 2 · 3 · 40 = 240 m² — the missing 9 m² is the little corner square that the square-of-a-sum formula remembers for you.
Convert it here: Area
Mechanics and braking energy
The energy that has to be shed braking from v₁ to v₂ is ½m(v₁² − v₂²) = ½m(v₁ − v₂)(v₁ + v₂). For a 1,000 kg car slowing from 20 m/s to 10 m/s that is 500 · 10 · 30 = 150,000 J — 150 kJ turned into heat in the brake pads.
Programming and cheaper arithmetic
Testing a² − b² = 0 over a million pairs costs two multiplications per pair — two million in total. Written as (a − b)(a + b) it costs one multiplication and two additions per pair: a million multiplications fewer, and an addition is markedly cheaper than a multiplication in a CPU.

All formulas

Square of a sum
$(a + b)^2 = a^2 + 2ab + b^2$
the middle term is twice the product, not the product
Square of a difference
$(a - b)^2 = a^2 - 2ab + b^2$
both squares positive, the minus sits on the middle term only
Difference of squares
$(a - b)(a + b) = a^2 - b^2$
the only one of these where the middle term disappears
Cube of a sum
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
coefficients 1, 3, 3, 1
Cube of a difference
$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
alternating signs, starting with a plus
Difference of cubes
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
a plus on the middle term of the bracket
Sum of cubes
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
a minus on the middle term of the bracket

Multiplying $(x + 7)(x + 7)$ can be done term by term and then collected. But the same computation comes back a dozen times a day and always ends the same way. Special products are those repeated results written down once — $(x + 7)^2$ becomes $x^2 + 14x + 49$ in a single move, with nothing to expand.

What matters more is that every one of them works both ways. Left to right it is expanding; right to left it is factoring, the step that opens cancelling in rational expressions and solving inequalities.

The square of a sum and of a difference

(a + b)^2 = a^2 + 2ab + b^2 \qquad (a - b)^2 = a^2 - 2ab + b^2

Where the middle term comes from is clearest as an area. A square of side $a + b$ cuts into four pieces: the square $a^2$ , the square $b^2$ and two identical rectangles of area $ab$ each.

A square of side a + b. Cut along the division of its side it leaves four pieces: a², b² and two rectangles a·b.

Two rectangles — hence $2ab$ , not $ab$ . That is the whole content of the identity, and the exact place where it is usually dropped.

In the square of a difference only the sign of the middle term changes. The outer terms are squares, so they stay positive:

(x - 5)^2 = x^2 - 10x + 25

Expand (2x + 3)² and (3a − 4b)².

The difference of squares

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

This is the only one of the identities where the middle term vanishes: expanding leaves $a^2 + ab - ba - b^2$ , and $ab$ cancels $ba$ .

Geometrically: cut a square of side $b$ out of a square of side $a$ , and rearrange what is left into a single rectangle with sides $a - b$ and $a + b$ .

A rectangle with sides a − b and a + b has exactly the area of the square a² with the square b² taken out.

The identity is the shortest route to a product done in the head. The numbers $98$ and $102$ sit symmetrically around a hundred, so:

98 \cdot 102 = (100 - 2)(100 + 2) = 100^2 - 2^2 = 10\,000 - 4 = 9996

A sum of squares $a^2 + b^2$ does not factor — there is no such identity, and the reason shows up in quadratic equations: $x^2 + 4 = 0$ has no real roots, so it has no real factors either.

Reading the identity backwards: factoring

The same equality read right to left turns a sum into a product:

x^2 - 9 = x^2 - 3^2 = (x - 3)(x + 3)

4x^2 + 20x + 25 = (2x)^2 + 2 \cdot 2x \cdot 5 + 5^2 = (2x + 5)^2

Recognizing a perfect square comes down to three checks: are the outer terms squares, do they share a sign, and is the middle term twice the product of their roots. If any one of them fails, the identity is not there.

Factor 49 − 25y² and x² + 12x + 36.

Cubes

The same identities one degree up. The cube of a sum has coefficients $1, 3, 3, 1$ , and the cube of a difference has the same coefficients with alternating signs:

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

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

Two further identities factor a sum and a difference of cubes. Unlike squares, both of them factor here:

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

The signs are crossed, and that is the only thing to remember: the minus in the first bracket goes with a plus on the middle term of the second, and the other way round.

x^3 - 8 = x^3 - 2^3 = (x - 2)(x^2 + 2x + 4)

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

expand (x − 1)²

Common mistakes

Losing the doubling — $(a + b)^2$ is $a^2 + 2ab + b^2$ , never $a^2 + b^2$ . The numerical check takes three seconds: $(3 + 4)^2 = 49$ , while $9 + 16 = 25$ .
Not squaring the coefficient — in $(2x + 3)^2$ the first term is $(2x)^2 = 4x^2$ , not $2x^2$ .
A negative square — in $(a - b)^2$ the minus belongs to the middle term only; $b^2$ stays positive, because it is a square.
Factoring a sum of squares — $a^2 + b^2$ has no factorization over the reals. Only the difference factors.
Crossing the signs in the cubes — in the factorization of $a^3 - b^3$ the second bracket carries a plus on $ab$ , even though the original difference carries a minus.

Formula card

Topic: Special products

Square of a sum
$(a + b)^2 = a^2 + 2ab + b^2$
the middle term is twice the product, not the product
Square of a difference
$(a - b)^2 = a^2 - 2ab + b^2$
both squares positive, the minus sits on the middle term only
Difference of squares
$(a - b)(a + b) = a^2 - b^2$
the only one of these where the middle term disappears
Cube of a sum
$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
coefficients 1, 3, 3, 1
Cube of a difference
$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
alternating signs, starting with a plus
Difference of cubes
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
a plus on the middle term of the bracket
Sum of cubes
$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
a minus on the middle term of the bracket

A square of side a + b. Its area is (a + b)², and cut apart it is a², b² and two rectangles a·b.

A rectangle with sides a − b and a + b comes from the square a² with the square b² cut out — which is why its area is a² − b².

Frequently asked questions

A balance scale in perfect equilibrium rendered as a geometric form, with a straight line crossing an axis like an orbital path, against a cosmic nebula in violet and magenta

Algebra

What x really means — unknown, variable and parameter

The letter x does not always mean the same thing. Sometimes it is a number you are hunting for, sometimes a quantity that varies continuously, sometimes a blank waiting for any number at all. Here is where algebraic notation came from, why it took two thousand years to arrive, what exactly separates an unknown from a variable and a parameter, and why an inequality flips when you multiply by a negative number.

11 min