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Perfect Square Trinomial: Definition, Formula & Examples

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  • 0:00 Perfect Square Trinomials
  • 1:11 The Square of a Binomial
  • 2:41 Factoring
  • 3:52 Special Products
  • 4:21 Completing the Square
  • 5:33 Lesson Summary
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Lesson Transcript
Instructor: David Karsner
Perfect square trinomials are a special group of polynomials that can be factored into a very convenient pattern, making them very useful in solving equations.

Perfect Square Trinomials

Before we can get to defining a perfect square trinomial, we need to review some vocabulary.

Perfect squares are numbers or expressions that are the product of a number or expression multiplied to itself. 7 times 7 is 49, so 49 is a perfect square. x squared times x squared equals x to the fourth, so x to the fourth is a perfect square.

  • Binomials are algrebraic expressions containing only two terms. Example: x + 3
  • Trinomials are algebraic expressions that contain three terms. Example: 3x2 + 5x - 6

Perfect square trinomials are algebraic expressions with three terms that are created by multiplying a binomial to itself. Example: (3x + 2y)2 = 9x2 + 12xy + 4y2

Recognizing when you have these perfect square trinomials will make factoring them much simpler. They are also very helpful when solving and graphing certain kinds of equations.

The Square of a Binomial

With perfect square trinomials, you will need to be able to move forwards and backwards. You should be able to take the binomials and find the perfect square trinomial and you should be able to take the perfect square trinomials and create the binomials from which it came. Any time you take a binomial and multiply it to itself, you end up with a perfect square trinomial. For example, take the binomial (x + 2) and multiply it by itself (x + 2).

(x + 2)(x + 2) = x2 + 4x + 4

The result is a perfect square trinomial.

To find the perfect square trinomial from the binomial, you will follow four steps:

Step One: Square the a

Step Two: Square the b

Step Three: Multiply 2 by a by b

Step Four: Add a2, b2, and 2ab

(a + b)2 = a2 + 2ab + b2

Let's add some numbers now and find the perfect square trinomial for 2x - 3y. For this:

a = 2x
b = 3y

Step One: Square the a
a2 = 4x2

Step Two: Square the b
b2 = 9y2

Step Three: Multiply 2 by a by 'b
2(2x)(-3y) = -12xy

Step Four: Add a2, b2, and 2ab
4x2 - 12xy + 9y2

Factoring

A trinomial is a perfect square trinomial if it can be factored into a binomial multiplied to itself. (This is the part where you are moving the other way). In a perfect square trinomial, two of your terms will be perfect squares. If you do not have two perfect square terms, then this trinomial is not a perfect square trinomial.

Now you should find the square root of both perfect square terms. multiply the two square roots together and then by two. You should get the positive or negative version of the other term. Once again, if this is not the case, you do not have a perfect square trinomial.

For example, in the trinomial x2 - 12x + 36, both x2 and 36 are perfect squares.

The square root of x2 is x, the square root of 36 is 6, and 2 times x (which is the same as 1) times 6 equals 12x/-12x, which does equal the other term.

x2 - 12x + 36 can be factored into (x - 6)(x- 6), also written as (x - 6)2.

Special Products

Perfect square trinomials are often introduced in algebra courses in a section that would be entitled 'Special Products.' These polynomials are grouped this way because they have a unique pattern to factoring them. The difference of squares, the sum of cubes, and the difference of cubes are other polynomials that fall into the special products category. The unique pattern with perfect square trinomials is that their factors consist of the repetition of one binomial.

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