*Contributor: Mason Smith. Lesson ID: 11551*

It is nice to have shortcuts when squaring off with polynomials! Special cases have special shortcuts, and binomials that are perfect squares are perfect candidates. Learn the secret with a koi pond!

categories

subject

Math

learning style

Visual

personality style

Lion

Grade Level

High School (9-12)

Lesson Type

Quick Query

Parent: What happened in math class today?

Student: When the teacher said to look for perfect squares, everyone looked at me.

When working with binomials, there are special cases when you need to find the product, but you don't need to use the FOIL (**f**irst, **o**uter, **i**nner, **l**ast) method.

- When is this, you may ask?

When you have a *perfect square*.

Imagine a square with sides of length (a + b).

The area of the square is described as (a + b)(a + b) or (a + b)^{2}. The area can also be found by adding the areas of the smaller squares and the rectangles inside. The sum of the areas is a^{2} + ab + ab +a + b^{2}.

This means that (a + b)^{2} = a^{2} + 2ab + b^{2}.

A trinomial of the form a^{2} + 2ab + b^{2} is called a perfect-square trinomial. A perfect-square trinomial is the result of squaring a binomial.

Finding the products of a perfect square trinomial is much easier than FOILing.

Look at some examples.

- (x + 4)
^{2}- Use the rule for (a + b)
^{2}- (a + b)
^{2}= a^{2}+ 2ab + b^{2}

- (a + b)
- Identify a and b; a = x and b = 4
- x
^{2}+ 2(x)(4) + 4^{2}

- x
- Simply
- x
^{2}+ 8x + 16

- x

- Use the rule for (a + b)

- (3x + 2y)
^{2}- Use the rule for (a + b)
^{2}- (a + b)
^{2}= a^{2}+ 2ab + b^{2}

- (a + b)
- Identify a and b; a = 3x and b = 2y
- (3x)
^{2}+ 2(3x)(2y) + (2y)^{2}

- (3x)
- Simply
- 9x
^{2}+ 12xy + 4y^{2}

- 9x

- Use the rule for (a + b)

- (4 + s
^{2})^{2}- Use the rule for (a + b)
^{2}- (a + b)
^{2}= a^{2}+ 2ab + b^{2}

- (a + b)
- Identify a and b; a = 4 and b = s
^{2}- 4
^{2}+ 2(4)(s^{2}) + (s^{2})^{2}

- 4
- Simply
- 16 + 8s
^{2}+ s^{4}

- 16 + 8s

- Use the rule for (a + b)

- (-m + 3)
^{2}- Use the rule for (a + b)
^{2}- ( a + b)
^{2}= a^{2}+ 2ab + b^{2}

- ( a + b)
- Identify a and b; a = m and b = 3
- (-m)
^{2}+ 2(-m)(3) + 3^{2}

- (-m)
- Simply
- m
^{2}- 6x + 9

- m

- Use the rule for (a + b)

Try these problems on your own. Answers to all problems from this lesson can be found in the *Special Products of Binomials Answer Key* located in **Downloadable Resources** in the right-hand sidebar (No peeking!).

Multiply

- (x + 6)
^{2} - (5a + b)
^{2} - (1 + c
^{3})^{2}

A *trinomial* of the form a^{2 }- 2ab + b^{2} is also a perfect square trinomial because it results from squaring the binomial (a - b).

Note: The sign in the binomial that is being squared makes a difference in the signs of the trinomial!

Walk through some examples.

- (x - 5)
^{2}- Use the rule for (a - b)
^{2}- (a - b)
^{2}= a^{2}- 2ab + b^{2}

- (a - b)
- Identify a and b; a = x and b = 5
- x
^{2}- 2(x)(5) + 5^{2}

- x
- Simply
- x
^{2}- 10x + 25

- x

- Use the rule for (a - b)

- (6a - 1)
^{2}- Use the rule for (a - b)
^{2}- (a - b)
^{2}= a^{2}- 2ab + b^{2}

- (a - b)
- Identify a and b; a = 6a and b = 1
- (6a)
^{2}- 2(6a)(1) + 1^{2}

- (6a)
- Simply
- 36a
^{2}- 12a + 1

- 36a

- Use the rule for (a - b)

- (4c - 6d)
^{2}- Use the rule for (a - b)
^{2}- (a - b)
^{2}= a^{2}- 2ab + b^{2}

- (a - b)
- Identify a and b; a = 4c and b = 3d
- (4c)
^{2}- 2(4c)(3d) + (3d)^{2}

- (4c)
- Simply
- 16c
^{2}- 24cd + 9c^{2}

- 16c

- Use the rule for (a - b)

- (3 - x
^{2})^{2}- Use the rule for (a - b)
^{2}- ( a - b)
^{2}= a^{2}- 2ab + b^{2}

- ( a - b)
- Identify a and b; a = 3 and b = x
^{2}- 3
^{2}- 2(3)(x^{2}) + (x^{2})^{2}

- 3
- Simply
- 9 - 6x
^{2}+ x^{4}

- 9 - 6x

- Use the rule for (a - b)

Try these problems (and check your answers when ready).

Multiply

- (x - 7)
^{2} - (3b - 2c)
^{2} - (a
^{2 }- 4)^{2}

One more special product of binomials is called the *difference of two squares*. This has the formula (a + b)(a - b) = a^{2 }- b^{2}.

Try out a few examples.

- (x +6)(x - 6)
- Use the rule for the difference of squares.
- (a + b)(a - b) = a
^{2}- b^{2}

- (a + b)(a - b) = a
- Identify a and b; a = x and b = 6.
- x
^{2}- 6^{2}

- x
- Simply.
- x
^{2}- 36

- x

- Use the rule for the difference of squares.

- (x
^{2}+ 2y)(x^{2}- 2y)- Use the rule for the difference of squares.
- (a + b)(a - b) = a
^{2}- b^{2}

- (a + b)(a - b) = a
- Identify a and b; a = x
^{2 }and b = 2y.- (x
^{2})^{2}- (2y)^{2}

- (x
- Simply.
- x
^{4}- 4y^{2}

- x

- Use the rule for the difference of squares.

- (7 + n)(7 - n)
- Use the rule for the difference of squares.
- (a + b)(a - b) = a
^{2}- b^{2}

- (a + b)(a - b) = a
- Identify a and b; a = 7 and b = n.
- 7
^{2}- n^{2}

- 7
- Simply.
- 49 - n
^{2}

- 49 - n

- Use the rule for the difference of squares.

Try these problems (and check your work).

Multiply

- (x + 8)(x - 8)
- (3 + 2y
^{2})(3 - 2y^{2}) - (9 + r)(9 - r)

Now that you have been introduced to these shortcuts, practice them in the *Got It?* section.