Special Products of Binomials

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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.

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

NameFormulaExample
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

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

Worked Examples

Example 1: Square of a sum

Multiply (x + 6)^2.

Here a = x and b = 6.

\begin{aligned} (x + 6)^2 &= x^2 + 2 \cdot x \cdot 6 + 6^2 \\ &= x^2 + 12x + 36 \end{aligned}

Check with x = 1: 7^2 = 49 and 1 + 12 + 36 = 49.

Example 2: Square of a difference

Multiply (y - 7)^2.

Here a = y and b = 7. The middle term is negative. The last term is positive, because (-7)^2 = 49.

\begin{aligned} (y - 7)^2 &= y^2 - 2 \cdot y \cdot 7 + 7^2 \\ &= y^2 - 14y + 49 \end{aligned}

Example 3: A coefficient in the first term

Multiply (3x + 2)^2.

Here a = 3x and b = 2. Square the whole first term, coefficient and all.

\begin{aligned} (3x + 2)^2 &= (3x)^2 + 2 \cdot 3x \cdot 2 + 2^2 \\ &= 9x^2 + 12x + 4 \end{aligned}

Check with FOIL: 9x^2 + 6x + 6x + 4 = 9x^2 + 12x + 4.

Example 4: Sum and difference

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

The two binomials have the same terms, 4x and 5y. One has a plus sign and one has a minus sign. So use a^2 - b^2.

\begin{aligned} (4x + 5y)(4x - 5y) &= (4x)^2 - (5y)^2 \\ &= 16x^2 - 25y^2 \end{aligned}

FOIL shows why there is no middle term. The outer product is -20xy and the inner product is +20xy. They cancel.

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.

\begin{aligned} (x + 3)^2 &= x^2 + 2 \cdot x \cdot 3 + 3^2 \\ &= x^2 + 6x + 9 \end{aligned}

The old area was x^2. So the added area is 6x + 9 square feet.

When x = 10, the new area is 100 + 60 + 9 = 169 square feet, and 13^2 = 169 checks it. The added area is 60 + 9 = 69 square feet.

Common Mistakes

MistakeExample of the errorFix
Forgetting the middle term(x + 5)^2 = x^2 + 25Square means times itself: x^2 + 10x + 25.
Using ab instead of 2ab(x + 3)^2 = x^2 + 3x + 9There 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 - 6Different 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.

ProblemRewriteAnswer
41 \cdot 39(40 + 1)(40 - 1) = 1600 - 11{,}599
51^2(50 + 1)^2 = 2500 + 100 + 12{,}601
29^2(30 - 1)^2 = 900 - 60 + 1841

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). If you see x^2 + 10x + 25, you will know it came from (x + 5)^2.

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

Practice Problems

  1. Multiply (x + 8)^2.
  2. Multiply (m - 3)^2.
  3. Multiply (x + 9)(x - 9).
  4. Multiply (2x + 5)^2.
  5. Multiply (3y - 4)^2.
  6. Multiply (6a + 1)(6a - 1).
  7. Multiply (x + 2y)^2.
  8. Multiply (5x - 2y)(5x + 2y).
  9. Use a special product to find 32 \cdot 28 without a calculator.
  10. A square photo has sides of x - 2 inches. Write its area as a polynomial. Then find the area when x = 10.
Show answers
  1. x^2 + 2 \cdot x \cdot 8 + 8^2 = x^2 + 16x + 64
  2. m^2 - 2 \cdot m \cdot 3 + 3^2 = m^2 - 6m + 9
  3. x^2 - 9^2 = x^2 - 81. The middle terms cancel.
  4. (2x)^2 + 2 \cdot 2x \cdot 5 + 5^2 = 4x^2 + 20x + 25
  5. (3y)^2 - 2 \cdot 3y \cdot 4 + 4^2 = 9y^2 - 24y + 16
  6. (6a)^2 - 1^2 = 36a^2 - 1
  7. x^2 + 2 \cdot x \cdot 2y + (2y)^2 = x^2 + 4xy + 4y^2
  8. (5x)^2 - (2y)^2 = 25x^2 - 4y^2. The order of the two binomials doesn’t matter.
  9. 32 \cdot 28 = (30 + 2)(30 - 2) = 900 - 4 = 896
  10. (x - 2)^2 = x^2 - 4x + 4 square inches. When x = 10: 100 - 40 + 4 = 64 square inches. Check: 8^2 = 64.

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.

Previous lesson: Multiplying Polynomials

Next lesson: Factoring Out the Greatest Common Factor

Back to the Algebra 1 Index

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