# Special Products

Special Products are the products of binomials that can be simplified further than regular products. A majority of these products can be expanded and simplified using the FOIL method covered in Expanding Polynomials.

## Distributive Law

Distributive Law involves products expressed in the form:

$$a(x + y)$$

These products can be expanded and simplified using the following process:

$$= (a)(x) + (a)(y)$$
$$= ax + ay$$

Example

Expand and simplify the expression $$5(a + 7)$$.

$$= (5)(a) + (5)(7)$$
$$= 5a + 35$$

Therefore, we can determine that $$5(a + 7)$$ expanded and simplified is $$5a + 35$$.

## Difference of 2 Squares

The Difference of Squares involves products expressed in the form:

$$(x + y)(x - y)$$

These products can be expanded and simplified using the following process:

$$= (x)(x) + (x)(-y) + (y)(x) + (y)(-y)$$
$$= x^2 - xy + xy - y^2$$
$$= x^2 - y^2$$
Example

Simplify the expression $$(6g + 7h)(6g - 7h)$$.

We can easily simplify the expression using the formula $$x^2 - y^2$$:

$$= (6g)^2 - (7h)^2$$
$$= 36g^2 - 49h^2$$

Therefore, we can determine that $$(6g + 7h)(6g - 7h)$$ simplified is $$36g^2 - 49h^2$$.

## Square of a Sum

The Square of a Sum involves products expressed in the form:

$$(x + y)^2$$

These products can be expanded and simplified using the following process:

$$= (x + y)(x + y)$$
$$= (x)(x) + (x)(y) + (y)(x) + (y)(y)$$
$$= x^2 + 2xy + y^2$$

Example

Expand and simplify the expression $$(4t + 3s)^2$$.

We can easily simplify the expression using the formula $$x^2 + 2xy + y^2$$:

$$= (4t)^2 + 2(4t)(3s) + (3s)^2$$
$$= 16t^2 + 24st + 9s^2$$

Therefore, we can determine that $$(4t + 3s)^2$$ expanded and simplified is $$16t^2 + 24st + 9s^2$$.

## Square of a Difference

The Square of a Difference involves products expressed in the form:

$$(x - y)^2$$

These products can be expanded and simplified using the following process:

$$= (x - y)(x - y)$$
$$= (x)(x) + (x)(-y) + (-y)(x) + (-y)(-y))$$
$$= x^2 -2xy + y^2$$

Example

Expand and simplify the expression $$(5q - 8)^2$$.

We can easily simplify the expression using the formula $$x^2 -2xy + y^2$$:

$$= (5q)^2 + (2)(5q)(8) + (8)^2$$
$$= 25q^2 - 80q + 64$$

Therefore, we can determine that $$(5q - 8)^2$$ expanded and simplified is $$25q^2 - 80q + 64$$.

Expand and simplify the following expressions:

$$(5x^2 + 7y^2)^2$$

$$(x - 3)^2 - (x + 3)(x - 3)$$

$$(3x^2 + 5x - 1)^2$$

$$(2x - 3)^3$$

The side length of a square is represented by $$x\;[cm]$$. The length of a rectangle is $$3\;[cm]$$ greater than the side length of the square. The width of the rectangle is $$3\;[cm]$$ less than the side length of the square. Which figure has the greater area and by how much?