Special products and factoring formulas

The formulas#

The square of a sum and the square of a difference:

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

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

The difference of squares:

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

The cube of a sum and the cube of a difference:

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

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

The sum and difference of cubes:

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

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

Notice two things. In the square of a difference the last term has a plus sign: minus times minus gives plus. And in the cube formulas the second bracket has no 2 in front of the product, and the sign inside it is the opposite of the sign between the cubes.

Worked examples: expanding brackets#

First example — $(x+3)^2$.

Step 1. Write the square of the first term: $x^2$. The first term here is $x$, so its square is a letter too.

Step 2. Add twice the product: $2 \cdot x \cdot 3 = 6x$. The 2 comes from the formula; you do not need to look for it in the problem.

Step 3. Add the square of the second term: $3^2 = 9$.

Answer: $x^2 + 6x + 9$.

Second example — $(2x-5)^2$. Here the first term is not a single letter but a product, and that is the main trap.

Step 1. The square of the first term is the square of the whole $2x$: $(2x)^2 = 4x^2$. The 2 gets squared as well, not just the letter.

Step 2. Twice the product: $2 \cdot 2x \cdot 5 = 20x$. The formula for a difference has a minus between the squares, so this term is subtracted.

Step 3. The square of the second term: $5^2 = 25$, with a plus sign.

Answer: $4x^2 - 20x + 25$.

Worked example: factoring#

You need the reverse direction more often than the forward one: it simplifies fractions and solves equations. Take $49y^2 - 16$.

Step 1. Check that both terms are perfect squares: $49y^2 = (7y)^2$ and $16 = 4^2$. There is a minus between them, so the difference of squares applies.

Step 2. Write it out by the formula: $(7y-4)(7y+4)$.

Step 3. Check by multiplying back: $7y \cdot 7y = 49y^2$, the middle terms $-28y$ and $+28y$ cancel, and $-16$ is left. It matches.

An important limit: a sum of squares $a^2 + b^2$ does not factor over the real numbers. If there is a plus, you cannot use the difference of squares — this is not a "forgotten sign", it is a different expression.

Mental arithmetic without a calculator#

The formulas work on ordinary numbers too, if you split them into a round number and a remainder.

$101^2 = (100+1)^2 = 10000 + 200 + 1 = 10201$.

$98 \cdot 102 = (100-2)(100+2) = 10000 - 4 = 9996$.

$41^2 - 39^2 = (41-39)(41+39) = 2 \cdot 80 = 160$. Squaring each number separately takes about three times as long and gives the same answer: $1681 - 1521 = 160$.

This trick is also a good self-check. If the mental answer from the formula matches the long multiplication, you applied the formula correctly.

Where people go wrong#

First place, by a wide margin, goes to the missing middle term: writing $(a+b)^2 = a^2 + b^2$. You can test yourself with numbers: put $a=2$, $b=3$. The left side is $(2+3)^2 = 25$, the right side is $4 + 9 = 13$. The numbers do not match, so the equation is false. Do this substitution every time you doubt a formula: it takes ten seconds and gives a clear answer.

The second mistake is the sign in the square of a difference. The last term always has a plus sign; only the middle term gets the minus.

The third is squaring only part of a term: $(3x)^2$ is $9x^2$, not $3x^2$. This rule is covered in detail in laws of exponents.

The fourth is using a formula on something that does not fit it. Before you write the answer, say out loud what your $a$ is and what your $b$ is. If you cannot name them, it is the wrong formula.

Factoring is most often needed to simplify a fraction or solve an equation. When factoring leaves you with an equation like $5x = 30$, the usual rules from how to solve linear equations take over.

What to do next#

Write the seven formulas on one sheet and keep it in front of you for the first week. Every day expand five brackets and factor five expressions back, always checking by substituting a number. The topic is done when you can tell which formula an expression needs just by looking at it, and the working takes less than a minute.

Step-by-step plan

1. Step 1 — learn the first three formulasSquare of a sum, square of a difference, difference of squares. Check each one by substituting a=2, b=3.
2. Step 2 — expand bracketsTwenty expressions like (x+5)^2 and (3a-2)^2, each in three lines: square, twice the product, square.
3. Step 3 — the reverse directionFactor expressions using the difference of squares and check by multiplying back.
4. Step 4 — cubesAdd the cube formulas; tell them apart from the square ones by the powers of the terms.
5. Step 5 — mental mathWork out 99², 102 · 98 and 51² − 49² with the formulas and compare with long multiplication.

Check yourself

Sources

• Khan Academy — Algebra 1Free practice on special products and factoring with instant feedback
  free
• OpenStax Elementary Algebra 2eFree textbook; the polynomials and factoring chapters cover all these formulas
  free

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