Special Products of Binomials: Formulas

Special products of binomials are shortcuts for three products that show up all the time. Squaring a sum gives $(a + b)^2 = a^2 + 2ab + b^2$ . Squaring a difference gives $(a - b)^2 = a^2 - 2ab + b^2$ . A sum times a difference gives $(a + b)(a - b) = a^2 - b^2$ . Learn the patterns and you can skip FOIL.

Before this lesson: You should know multiplying binomials, the exponent rules, and combining like terms.

Extra practice: The Algebra 1 Unit 7 Practice Pack (Part 1) has printable problems, answer keys and a cheat sheet for this unit.

Why Special Products Have Patterns

Some binomial products have the same two terms in both binomials. When that happens, the answer always follows a pattern. You can still use FOIL, but the pattern is faster.

Look at $(x + 4)^2$ . It means $(x + 4)(x + 4)$ . Draw it as a square with sides $x + 4$ . The square splits into four parts: $x^2$ , two rectangles of $4x$ , and $16$ .

A square with each side split into x and 4. It has four parts: a square labeled x², two rectangles each labeled 4x, and a small square labeled 16. Beside it: (x + 4)² = x² + 4x + 4x + 16 = x² + 8x + 16. — The two middle rectangles are both 4x. That is why the middle term is 2 · 4x = 8x.

The two middle rectangles are the same size. So the middle term is always 2 times the product of the two terms. That is where the 2 in the pattern comes from.

Special Products of Binomials: The Three Formulas

Name	Formula	Example
Square of a sum	$(a + b)^2 = a^2 + 2ab + b^2$	$(x + 3)^2 = x^2 + 6x + 9$
Square of a difference	$(a - b)^2 = a^2 - 2ab + b^2$	$(x - 3)^2 = x^2 - 6x + 9$
Sum and difference	$(a + b)(a - b) = a^2 - b^2$	$(x + 3)(x - 3) = x^2 - 9$

In each formula, $a$ is the first term and $b$ is the second term. They can be numbers, variables or terms like $3x$ .

The first two answers are called perfect square trinomials. A trinomial is a polynomial with three terms. The last answer is called a difference of squares.

Steps

Check the pattern. Is it a binomial squared? Or the same two terms with a plus sign and a minus sign?
Name $a$ and $b$ . Include any coefficient, like $a = 3x$ .
Use the formula that matches.
Square each whole term. For example, $(3x)^2 = 9x^2$ .
Check with FOIL or by putting in a number, like $x = 1$ .

Worked Examples

Example 1: Square of a sum

Multiply $(x + 6)^2$ .

Step 1. Match it to the pattern $(a + b)^2 = a^2 + 2ab + b^2$ . Here $a = x$ and $b = 6$ .

(x + 6)^2 = x^2 + 2 \cdot x \cdot 6 + 6^2

Step 2. Substitute into the pattern.

= x^2 + 12x + 6^2

Step 3. Multiply the middle term: $2 \cdot x \cdot 6 = 12x$ .

= x^2 + 12x + 36

Step 4. Square the last term: $6^2 = 36$ .

Answer: $x^2 + 12x + 36$

Check: Use $x = 1$ . The original gives $(1 + 6)^2 = 7^2 = 49$ . The answer gives $1 + 12 + 36 = 49$ .

Example 2: Square of a difference

Multiply $(y - 7)^2$ .

Step 1. Match it to the pattern $(a - b)^2 = a^2 - 2ab + b^2$ . Here $a = y$ and $b = 7$ .

(y - 7)^2 = y^2 - 2 \cdot y \cdot 7 + 7^2

Step 2. Substitute into the pattern. The middle term gets the minus sign.

= y^2 - 14y + 7^2

Step 3. Multiply the middle term: $2 \cdot y \cdot 7 = 14y$ .

= y^2 - 14y + 49

Step 4. Square the last term. It is positive, because $(-7)^2 = 49$ too (see multiplying and dividing integers).

Answer: $y^2 - 14y + 49$

Example 3: A coefficient in the first term

Multiply $(3x + 2)^2$ .

Step 1. Match it to $(a + b)^2 = a^2 + 2ab + b^2$ . Here $a = 3x$ and $b = 2$ .

(3x + 2)^2 = (3x)^2 + 2 \cdot 3x \cdot 2 + 2^2

Step 2. Substitute into the pattern. Keep 3x in parentheses so the whole term gets squared.

= 9x^2 + 2 \cdot 3x \cdot 2 + 2^2

Step 3. Square the first term, coefficient and all: $(3x)^2 = 9x^2$ (see exponent rules).

= 9x^2 + 12x + 2^2

Step 4. Multiply the middle term: $2 \cdot 3x \cdot 2 = 12x$ .

= 9x^2 + 12x + 4

Step 5. Square the last term.

Answer: $9x^2 + 12x + 4$

Check: Use FOIL (see multiplying binomials): $(3x + 2)(3x + 2) = 9x^2 + 6x + 6x + 4 = 9x^2 + 12x + 4$ .

Example 4: Sum and difference

Multiply $(4x + 5y)(4x - 5y)$ .

Step 1. The two binomials have the same terms, $4x$ and $5y$ . One has a plus sign and one has a minus sign. So use $(a + b)(a - b) = a^2 - b^2$ with $a = 4x$ and $b = 5y$ .

(4x + 5y)(4x - 5y) = (4x)^2 - (5y)^2

Step 2. Substitute into the pattern.

= 16x^2 - (5y)^2

Step 3. Square the first term.

= 16x^2 - 25y^2

Step 4. Square the second term.

Answer: $16x^2 - 25y^2$

Check: FOIL shows why there is no middle term. The outer product is $4x(-5y) = -20xy$ and the inner product is $5y \cdot 4x = 20xy$ . They add to 0.

Example 5: Word problem

A square garden has sides of $x$ feet. The owner adds 3 feet to each side. Write the new area as a polynomial. How much area was added? Find both when $x = 10$ .

A = (x + 3)^2

Step 1. Each new side is $x + 3$ feet. The area of a square is side times side (see area of rectangles).

= x^2 + 2 \cdot x \cdot 3 + 3^2

Step 2. Use the square of a sum with $a = x$ and $b = 3$ .

= x^2 + 6x + 3^2

Step 3. Multiply the middle term.

= x^2 + 6x + 9

Step 4. Square the last term.

(x^2 + 6x + 9) - x^2 = 6x + 9

Step 5. The old area was $x^2$ . Subtract it to find the added area.

10^2 + 6(10) + 9 = 100 + 60 + 9 = 169

Step 6. Substitute 10 for x in the new area.

6(10) + 9 = 60 + 9 = 69

Step 7. Substitute 10 for x in the added area.

Answer: The new area is $x^2 + 6x + 9$ square feet, and $6x + 9$ square feet were added. When $x = 10$ , the new area is 169 square feet and 69 square feet were added.

Check: The new garden is 13 feet on a side, and $13^2 = 169$ .

Common Mistakes

Mistake	Example of the error	Fix
Forgetting the middle term	$(x + 5)^2 = x^2 + 25$	Square means times itself: $x^2 + 10x + 25$ .
Using ab instead of 2ab	$(x + 3)^2 = x^2 + 3x + 9$	There are two middle rectangles: $x^2 + 6x + 9$ .
Making the last term negative	$(x - 4)^2 = x^2 - 8x - 16$	$(-4)^2$ is positive: $x^2 - 8x + 16$ .
Not squaring the coefficient	$(3x + 1)^2 = 3x^2 + 6x + 1$	$(3x)^2 = 9x^2$ , so $9x^2 + 6x + 1$ .
Using the pattern when the terms don’t match	$(x + 2)(x - 3) = x^2 - 6$	Different numbers, so use FOIL: $x^2 - x - 6$ .

Using Special Products

Mental math

The patterns also work with plain numbers. Write a number as a sum or difference that is easy to square.

Problem	Rewrite	Answer
$41 \cdot 39$	$(40 + 1)(40 - 1) = 1600 - 1$	$1{,}599$
$51^2$	$(50 + 1)^2 = 2500 + 100 + 1$	$2{,}601$
$29^2$	$(30 - 1)^2 = 900 - 60 + 1$	$841$

Getting ready to factor

Later in this unit, you will run these patterns backward. If you see $x^2 - 49$ , you will know it came from $(x + 7)(x - 7)$ . See factoring the difference of squares. If you see $x^2 + 10x + 25$ , you will know it came from $(x + 5)^2$ . See perfect square trinomials.

For a binomial raised to a higher power, like $(x + 1)^3$ , multiply step by step. See multiplying polynomials.

Practice Problems

Multiply $(x + 8)^2$ .
Multiply $(m - 3)^2$ .
Multiply $(x + 9)(x - 9)$ .
Multiply $(2x + 5)^2$ .
Multiply $(3y - 4)^2$ .
Multiply $(6a + 1)(6a - 1)$ .
Multiply $(x + 2y)^2$ .
Multiply $(5x - 2y)(5x + 2y)$ .
Use a special product to find $32 \cdot 28$ without a calculator.
A square photo has sides of $x - 2$ inches. Write its area as a polynomial. Then find the area when $x = 10$ .

Show answers

Use $(a + b)^2 = a^2 + 2ab + b^2$ with $a = x$ and $b = 8$

x^2 + 2 \cdot x \cdot 8 + 8^2

Substitute.

x^2 + 16x + 8^2

Multiply the middle term.

x^2 + 16x + 64

Square the last term.

Use $(a - b)^2 = a^2 - 2ab + b^2$ with $a = m$ and $b = 3$

m^2 - 2 \cdot m \cdot 3 + 3^2

Substitute.

m^2 - 6m + 3^2

Multiply the middle term.

m^2 - 6m + 9

Square the last term.

Use $(a + b)(a - b) = a^2 - b^2$ with $a = x$ and $b = 9$

x^2 - 9^2

Substitute.

x^2 - 81

Square 9.

The middle terms cancel.

Use the square of a sum with $a = 2x$ and $b = 5$

(2x)^2 + 2 \cdot 2x \cdot 5 + 5^2

Substitute.

4x^2 + 2 \cdot 2x \cdot 5 + 5^2

Square the first term.

4x^2 + 20x + 5^2

Multiply the middle term.

4x^2 + 20x + 25

Square the last term.

Use the square of a difference with $a = 3y$ and $b = 4$

(3y)^2 - 2 \cdot 3y \cdot 4 + 4^2

Substitute.

9y^2 - 2 \cdot 3y \cdot 4 + 4^2

Square the first term.

9y^2 - 24y + 4^2

Multiply the middle term.

9y^2 - 24y + 16

Square the last term.

Use $(a + b)(a - b) = a^2 - b^2$ with $6a$ and 1

(6a)^2 - 1^2

Substitute.

36a^2 - 1^2

Square the first term.

36a^2 - 1

Square 1.

Use the square of a sum with $a = x$ and $b = 2y$

x^2 + 2 \cdot x \cdot 2y + (2y)^2

Substitute.

x^2 + 4xy + (2y)^2

Multiply the middle term.

x^2 + 4xy + 4y^2

Square the last term.

It is a sum times a difference of the same two terms. The order of the binomials doesn’t matter

(5x)^2 - (2y)^2

Substitute into the pattern.

25x^2 - (2y)^2

Square the first term.

25x^2 - 4y^2

Square the second term.

32 \cdot 28 = (30 + 2)(30 - 2)

Write each number as 30 plus or minus 2.

30^2 - 2^2

Use the sum-and-difference pattern.

900 - 4

Square each number.

896

Subtract.

10.

A = (x - 2)^2

The area of a square is side times side.

x^2 - 2 \cdot x \cdot 2 + 2^2

Use the square of a difference.

x^2 - 4x + 2^2

Multiply the middle term.

x^2 - 4x + 4

Square the last term.

10^2 - 4(10) + 4 = 100 - 40 + 4 = 64

Substitute 10 for x.

Check: the photo is 8 inches on a side: $8^2 = 64$
The area is 64 square inches.

Want more practice? The Algebra 1 Unit 7 Practice Pack (Part 1) has 37 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

FAQ

What are the three special products?

They are the square of a sum, $(a + b)^2$ , the square of a difference, $(a - b)^2$ , and a sum times a difference, $(a + b)(a - b)$ .

What is the formula for squaring a binomial?

Square the first term, add 2 times the product of the terms, then add the square of the last term: $(a + b)^2 = a^2 + 2ab + b^2$ . For a difference, the middle term is subtracted.

Why does (a + b)(a − b) have no middle term?

The outer product is $-ab$ and the inner product is $+ab$ . They add to 0, so only $a^2 - b^2$ is left.

Is (a + b)² the same as a² + b²?

No. You must include the middle term $2ab$ . For example, $(2 + 3)^2 = 25$ , but $2^2 + 3^2 = 13$ .

Summary

Square of a sum: $(a + b)^2 = a^2 + 2ab + b^2$ .
Square of a difference: $(a - b)^2 = a^2 - 2ab + b^2$ . The last term is still positive.
Sum and difference: $(a + b)(a - b) = a^2 - b^2$ . The middle terms cancel.
Square the whole term, including its coefficient.
If the terms don’t match, use FOIL instead.

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Next lesson: Factoring Out the Greatest Common Factor

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