*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

Middle School (6-8), 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.

(If you are not familiar with the FOIL method, please go to the previous *Polynomials* lessons, found under **Related Lessons** in the right-hand sidebar.)

- 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 are can also be found by adding the ares 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. Let's do 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 and discuss your answers with a parent or teacher. 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 is the result of squaring the binomial (a - b).

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

Let's 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 discuss your answers with a parent or teacher.

Multiply:

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

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

Let's try out a few examples.

- (x +6)(x - 6)
- Use the rule for 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 difference of squares
- (x
^{2}+ 2y)(x^{2}- 2y)- Use the rule for 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 difference of squares
- (7 + n)(7 - n)
- Use the rule for 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 difference of squares

Try these problems and discuss your answers with a parent or teacher.

Multiply:

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

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

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