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 involves products expressed in the form:
These products can be expanded and simplified using the following process:
\(= (a)(\textcolor{red}{x}) + (a)(\textcolor{blue}{y})\)
\(= a\textcolor{red}{x} + a\textcolor{blue}{y}\)
ExampleExpand and simplify the expression \(5(a + 7)\).
We can easily simplify the expression using Distributive Law where \(a = 5\), \(\textcolor{red}{x = a}\) and \(\textcolor{blue}{y = 7}\):
\(= a\textcolor{red}{x} + a\textcolor{blue}{y}\)
\(= (5)(\textcolor{red}{a}) + (5)(\textcolor{blue}{7})\)
\(= 5a + 35\)
Therefore, we can determine that \(5(a + 7)\) expanded and simplified is \(\boldsymbol{5a + 35}\).
The Difference of Squares involves products expressed in the form:
These products can be expanded and simplified using the following process:
\(= (\textcolor{red}{x})(\textcolor{red}{x}) + (\textcolor{red}{x})(-\textcolor{blue}{y}) + (\textcolor{blue}{y})(\textcolor{red}{x}) + (\textcolor{blue}{y})(-\textcolor{blue}{y})\)
\(= \textcolor{red}{x}^2 - \textcolor{red}{x}\textcolor{blue}{y} + \textcolor{red}{x}\textcolor{blue}{y} - \textcolor{blue}{y}^2\)
\(= \textcolor{red}{x}^2 - \textcolor{blue}{y}^2\)
ExampleSimplify the expression \((6g + 7h)(6g - 7h)\).
We can easily simplify the expression using the Difference of Squares where \(\textcolor{red}{x = 6g}\) and \(\textcolor{blue}{y = 7h}\):
\(\textcolor{red}{x}^2 - \textcolor{blue}{y}^2\)
\(= (\textcolor{red}{6g})^2 - (\textcolor{blue}{7h})^2\)
\(= 36g^2 - 49h^2\)
Therefore, we can determine that \((6g + 7h)(6g - 7h)\) simplified is \(\boldsymbol{36g^2 - 49h^2}\).
The Square of a Sum involves products expressed in the form:
These products can be expanded and simplified using the following process:
\(= (\textcolor{red}{x} + \textcolor{blue}{y})(\textcolor{red}{x} + \textcolor{blue}{y})\)
\(= (\textcolor{red}{x})(\textcolor{red}{x}) + (\textcolor{red}{x})(\textcolor{blue}{y}) + (\textcolor{blue}{y})(\textcolor{red}{x}) + (\textcolor{blue}{y})(\textcolor{blue}{y})\)
\(= \textcolor{red}{x}^2 + 2\textcolor{red}{x}\textcolor{blue}{y} + \textcolor{blue}{y}^2\)
ExampleExpand and simplify the expression \((4t + 3s)^2\).
We can easily simplify the expression using the Square of a Sum where \(\textcolor{red}{x = 4t}\) and \(\textcolor{blue}{y = 3s}\):
\(\textcolor{red}{x}^2 + 2\textcolor{red}{x}\textcolor{blue}{y} + \textcolor{blue}{y}^2\)
\(= (\textcolor{red}{4t})^2 + 2(\textcolor{red}{4t})(\textcolor{blue}{3s}) + (\textcolor{blue}{3s})^2\)
\(= 16t^2 + 24st + 9s^2\)
Therefore, we can determine that \((4t + 3s)^2\) expanded and simplified is \(\boldsymbol{16t^2 + 24st + 9s^2}\).
The Square of a Difference involves products expressed in the form:
These products can be expanded and simplified using the following process:
\(= (\textcolor{red}{x} - \textcolor{blue}{y})(\textcolor{red}{x} - \textcolor{blue}{y})\)
\(= (\textcolor{red}{x})(\textcolor{red}{x}) + (\textcolor{red}{x})(-\textcolor{blue}{y}) + (-\textcolor{blue}{y})(\textcolor{red}{x}) + (-\textcolor{blue}{y})(-\textcolor{blue}{y})\)
\(= \textcolor{red}{x}^2 -2\textcolor{red}{x}\textcolor{blue}{y} + \textcolor{blue}{y}^2\)
ExampleExpand and simplify the expression \((5q - 8)^2\).
We can easily simplify the expression using the Square of a Difference where \(\textcolor{red}{x = 5q}\) and \(\textcolor{blue}{y = 8}\):
\(\textcolor{red}{x}^2 -2\textcolor{red}{x}\textcolor{blue}{y} + \textcolor{blue}{y}^2\)
\(= (\textcolor{red}{5q})^2 - 2(\textcolor{red}{5q})(\textcolor{blue}{8}) + (\textcolor{blue}{8})^2\)
\(= 25q^2 - 80q + 64\)
Therefore, we can determine that \((5q - 8)^2\) expanded and simplified is \(\boldsymbol{25q^2 - 80q + 64}\).
\((5x^2 + 7y^2)^2\)
This expression is a Square of a Sum; therefore, we can expand and simplify the expression using its corresponding formula where \(\textcolor{red}{x = 5x^2}\) and \(\textcolor{blue}{y = 7y^2}\):
\(\textcolor{red}{x}^2 + 2\textcolor{red}{x}\textcolor{blue}{y} + \textcolor{blue}{y}^2\)
\(= (\textcolor{red}{5x^2})^2 + 2(\textcolor{red}{5x^2})(\textcolor{blue}{7y^2}) + (\textcolor{blue}{7y^2})^2\)
\(= 25x^4 + 70x^2y^2 + 49y^4\)
Therefore, we can determine that \((5x^2)^2 + 2(5x^2)(7y^2) + (7y^2)^2\) expanded and simplified is \(\boldsymbol{25x^4 + 70x^2y^2 + 49y^4}\).
\((x - 3)^2 - (x + 3)(x - 3)\)
This expression combines Square of a Difference and Difference of 2 Squares; therefore, we can combine their respective formulas where \(\textcolor{red}{x = x}\) and \(\textcolor{blue}{y = 3}\):
\(\textcolor{red}{x}^2 - 2\textcolor{red}{x}\textcolor{blue}{y} + \textcolor{blue}{y}^2\) - \(\textcolor{red}{x}^2 - \textcolor{blue}{y}^2\)
\(= [(\textcolor{red}{x})^2 - 2(\textcolor{red}{x})(\textcolor{blue}{3}) + (\textcolor{blue}{3})^2] - [(\textcolor{red}{x})^2 - (\textcolor{blue}{3})^2]\)
\(= x^2 - 6x + 9 - [x^2 - 9]\)
\(= x^2 - 6x + 9 - x^2 + 9\)
\(= -6x + 18\)
Therefore, we can determine that \((x - 3)^2 - (x + 3)(x - 3)\) expanded and simplified is \(\boldsymbol{-6x + 18}\).
\((3x^2 + 5x - 1)^2\)
SThis expression contains a trinomial instead of a binomial; therefore, we can expand the expression with \(9\) unique terms instead of the usual \(4\) where \(\textcolor{red}{x = 3x}\), \(\textcolor{blue}{y = 5x}\), and \(\textcolor{green}{z = -1}\):
\(= (\textcolor{red}{x} + \textcolor{blue}{y} - \textcolor{green}{z})(\textcolor{red}{x} + \textcolor{blue}{y} - \textcolor{green}{z})\)
\(= (\textcolor{red}{3x^2} + \textcolor{blue}{5x}\textcolor{green}{-1})(\textcolor{red}{3x^2} + \textcolor{blue}{5x}\textcolor{green}{-1})\)
\(= (3x^2)(3x^2) + (3x^2)(5x) + (3x^2)(-1) + (5x)(3x^2) + (5x)(5x) + (5x)(-1) + (-1)(3x^2) + (-1)(5x) + (-1)(-1)\)
\(= 9x^4 + 15x^3 - 3x^2 + 15x^3 + 25x^2 - 5x - 3x^2 - 5x + 1\)
\(= 9x^4 + 30x^3 + 19x^2 - 10x + 1\)
Therefore, we can determine that \((3x^2 + 5x - 1)^2\) expanded and simplified is \(\boldsymbol{9x^4 + 30x^3 + 19x^2 - 10x + 1}\).
\((2x - 3)^3\)
Since this expression is expanded by the power of \(3\) instead of \(2\), we consider this a Cube of a Difference; therefore, we can expand and simplify the expression using its corresponding formula where \(\textcolor{red}{x = 2x}\) and \(\textcolor{blue}{y = 3}\):
\(= \textcolor{red}{x}^3 - 3\textcolor{red}{x}^2\textcolor{blue}{y} + 3\textcolor{red}{x}\textcolor{blue}{y}^2 - \textcolor{blue}{y}^3\)
\(= (\textcolor{red}{2x})^3 - 3(\textcolor{red}{2x})^2(\textcolor{blue}{3}) + 3(\textcolor{red}{2x})(\textcolor{blue}{3})^2 - (\textcolor{blue}{3})^3\)
\(= 8x^3 - 9(4x^2) + 6x(9) - (27)\)
\(= 8x^3 - 36x^2 + 54x - 27\)
Therefore, we can determine that \((2x - 3)^3\) expanded and simplified is \(\boldsymbol{8x^3 - 36x^2 + 54x - 27}\).
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:
We can start by determining the Area of the Square:
\(A_{\text{Square}} = (x)(x)\)
\(A_{\text{Square}} = x^2\)
We can next determine the Area of the Rectangle using the Difference of \(2\) Squares:
\(A_{\text{Rectangle}} = (x + 3)(x - 3)\)
\(A_{\text{Rectangle}} = (x)(x) + (x)(-3) + (x)(3) + (3)(-3)\)
\(A_{\text{Rectangle}} = x^2 -3x + 3x - 9\)
\(A_{\text{Rectangle}} = x^2 - 9\)
Based on these measurements, we can determine that the square has a larger area than the rectangle.