Question

Create an algebraic expression for each of the following special products patterns. Use the special products formulas to complete the multiplication. Show all work for full credit.
A. perfect square trinomial formula
B. difference of perfect squares formula

Answer 1

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 + 3Trinomials 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 + 9y

Answer 2

A. Perfect Square Trinomial is an expression obtained from the square of a binomial equation. Any trinomial of the form $ax^2 + bx + c$ is said to a perfect square if it satisfies the condition $b^2 = 4ac$. Trinomial has two factors that are identical.

The perfect square trinomial formulas are:

$(ax+b)^2=(ax)^2+2(ax)\cdot b+b^2,\\ (ax-b)^2=(ax)^2-2(ax)\cdot b+b^2$.

For example,

$4x^2-12x+9=(2x)^2-2\cdot 2x\cdot 3+3^2=(2x-3)^2,\\ 36x^2+12x+1=(6x)^2+2\cdot 6x\cdot 1+1^2=(6x+1)^2$.

B. Difference of squares is the subtraction of two squares terms is a squared term minus from another squared term i.e. $a^2 - b^2$.

This may be factored according to the given below mathematical identity:

$a^2-b^2=(a-b)(a+b)$.

For example,

$2018^2-2017^2=(2018-2017)(2018+2017)=1\cdot 4035=4035,\\ 49m^2-81n^2=(7m)^2-(9n)^2=(7m-9n)(7m+9n)$.