**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:

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

Show Answer
This expression is a Square of a Sum; therefore, we can expand and simplify the expression using the formula \(x^2 + 2xy + y^2\):

\(= (5x^2)^2 + 2(5x^2)(7y^2) + (7y^2)^2\)

\(= 25x^4 + 70x^2y^2 + 49y^4\)
Therefore, we can determine that \((5x^2)^2 + 2(5x^2)(7y^2) + (7y^2)^2\) expanded and simplified is \(25x^4 + 70x^2y^2 + 49y^4\).

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

Show Answer
This expression combines Square of a Difference and Difference of 2 Squares; therefore, we can combine the formulas \(x^2 - 2xy + y^2\) and \(x^2 - y^2\):

\(= [(x)^2 - 2(x)(3) + (3)^2] - [(x)^2 - (3)^2]\)

\(= x^2 - 6x + 9 - [x^2 - 9]\)

\(= x^2 - 6x + 9 - x^2 + 9\)

\(= -6x + 18\)

Therefore, we can determine that \((x - 3)^2 - (x + 3)(x - 3)\) expanded and simplified is \(-6x + 18\).

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

Show Answer
Since this expression contains a trinomial instead of a binomial, we will be expanding the expression with 9 unique terms instead of the usual 4:

\(= (3x^2 + 5x - 1)(3x^2 + 5x - 1)\)

\(= (3x^2)(3x^2) + (3x^2)(5x) + (3x^2)(-1) + (5x)(3x^2) + (5x)(5x) + (5x)(-1) + (-1)(3x^2) + (-1)(5x) + (-1)(-1)\)

\(= 9x^4 + 15x^3 - 3x^2 + 15x^3 + 25x^2 - 5x - 3x^2 - 5x + 1\)

\(= 9x^4 + 30x^3 + 19x^2 - 10x + 1\)
Therefore, we can determine that \((3x^2 + 5x - 1)^2\) expanded and simplified is \(9x^4 + 30x^3 + 19x^2 - 10x + 1\)

\((2x - 3)^3\)

Show Answer
Since this expression is expanded by the power of 3 instead of 2, we consider this a Cube of a Difference, simplified in the form \(x^3 - 3x^2y + 3xy^2 - y^3\):

\(= (2x)^3 - 3(2x)^2(3) + 3(2x)(3)^2 - (3)^3\)

\(= 8x^3 - 9(4x^2) + 6x(9) - (27)\)

\(= 8x^3 - 36x^2 + 54x - 27\)
Therefore, we can determine that \((2x - 3)^3\) expanded and simplified is \(8x^3 - 36x^2 + 54x - 27\).

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?

Show Answer
First, we can draw sketches of the square and the rectangle to get a better idea of their respective measurements:

Now we can use the Area equation \(A = LW\) to determine their respective areas:

\(\text{Square} = (x)(x)\)

\(\text{Square} = x^2\)

\(\text{Rectangle} = (x + 3)(x - 3)\)

\(\text{Rectangle} = (x)(x) + (x)(-3) + (x)(3) + (3)(-3)\)

\(\text{Rectangle} = x^2 -3x + 3x - 9\)

\(\text{Rectangle} = x^2 - 9\)
Based on these measurements, we can determine that the square has a larger area than the rectangle.