Math Core

A few binomial products show up so often that it pays to know their results by heart. Recognizing these special products makes multiplying faster, and, just as important, it will let you spot them instantly when you factor in the last lesson of this unit.

Squaring a binomial

To square a binomial, multiply it by itself. Here is $(a + b)^2$ worked out once, in general:

\begin{aligned} (a + b)^2 &= (a + b)(a + b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 \end{aligned}

The two middle products, $ab$ and $ba$ , are equal, so they combine into $2ab$ . The same work with a minus sign gives $(a - b)^2 = a^2 - 2ab + b^2$ : the last term $(-b)(-b) = b^2$ is still positive, and only the middle term turns negative.

You can see $(a + b)^2$ as a picture. A square with side $a + b$ splits into a square of area $a^2$ , a square of area $b^2$ , and two rectangles of area $ab$ each.

$\times$	$a$	$b$
$a$	$a^2$	$ab$
$b$	$ab$	$b^2$

The product of a sum and a difference

Now multiply two binomials that differ only in the sign between the terms:

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

The outer and inner products are opposites, so they cancel, leaving no middle term at all. The result is called a difference of two squares.

Special product patterns

\begin{aligned} (a + b)^2 &= a^2 + 2ab + b^2 \\ (a - b)^2 &= a^2 - 2ab + b^2 \\ (a + b)(a - b) &= a^2 - b^2 \end{aligned}

In words: the square of a binomial is first squared, plus or minus twice the product, plus last squared. A sum times a difference is first squared minus last squared.

Common mistake

The most common mistake in all of algebra may be this one:

(x + 5)^2 \ne x^2 + 25.

Squaring does not distribute over addition. Test it with $x = 1$ : $(1 + 5)^2 = 36$ , but $1^2 + 25 = 26$ . The missing $10$ is the middle term $2 \cdot x \cdot 5 = 10x$ . Correct: $(x + 5)^2 = x^2 + 10x + 25$ .

Using the patterns

To use a pattern, decide what plays the role of $a$ and what plays the role of $b$ . Then substitute. When $a$ or $b$ has a coefficient, put it in parentheses so the whole term gets squared.

Worked example: Squares of binomials

Expand $(x + 5)^2$ and $(3y - 4)^2$ .

For $(x + 5)^2$ , use $a = x$ and $b = 5$ :

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

For $(3y - 4)^2$ , use $a = 3y$ and $b = 4$ , with a minus sign:

(3y - 4)^2 = (3y)^2 - 2(3y)(4) + 4^2 = 9y^2 - 24y + 16.

Notice $(3y)^2 = 9y^2$ , not $3y^2$ . The coefficient gets squared too.

Worked example: Sum times difference

Find $(2x + 7)(2x - 7)$ .

The binomials have the same terms, one with $+$ and one with $-$ . Use $a = 2x$ and $b = 7$ :

(2x + 7)(2x - 7) = (2x)^2 - 7^2 = 4x^2 - 49.

Worked example: Two variables

Expand $(5m - 2n)^2$ .

With $a = 5m$ and $b = 2n$ :

(5m - 2n)^2 = 25m^2 - 2(5m)(2n) + 4n^2 = 25m^2 - 20mn + 4n^2.

Mental math with special products

The patterns work for ordinary numbers too. Choose $a$ to be a round number close to the numbers you're multiplying.

Worked example: Multiplying in your head

Compute $48 \cdot 52$ and $31^2$ .

$48$ and $52$ are $50 - 2$ and $50 + 2$ , a sum and a difference:

48 \cdot 52 = (50 - 2)(50 + 2) = 2500 - 4 = 2496.

For $31^2$ , write $31 = 30 + 1$ :

31^2 = 30^2 + 2(30)(1) + 1^2 = 900 + 60 + 1 = 961.

Tip

A quick check for a squared binomial: the first and last terms of the answer must both be perfect squares and both be positive, and the middle term's coefficient must be twice the product of their square roots. For $9y^2 - 24y + 16$ : $\sqrt{9y^2} = 3y$ , $\sqrt{16} = 4$ , and $2 \cdot 3 \cdot 4 = 24$ . ✓

Practice

Practice 1

Expand $(x + 6)^2$ .

Practice 2

When $(x - 9)^2$ is expanded, what is the coefficient of $x$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

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

Practice 4

Find $(x + 8)(x - 8)$ .

Practice 5

When $(4m + 3n)^2$ is expanded, what is the coefficient of $mn$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Use a special product to compute $39 \cdot 41$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

The expression $(x + k)^2$ expands to $x^2 + 14x + 49$ . What is $k$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

Find $(x^2 + 3)(x^2 - 3)$ .