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:
\((5x^2 + 7y^2)^2\)
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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 \(25x^4 + 70x^2y^2 + 49y^4\).
\((x - 3)^2 - (x + 3)(x - 3)\)
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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 \(-6x + 18\).
\((3x^2 + 5x - 1)^2\)
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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 \(9x^4 + 30x^3 + 19x^2 - 10x + 1\)
\((2x - 3)^3\)
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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?
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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.