The Difference of Squares is a way of showing how an expression in the form \(x^2 - y^2\) can be written as the product of 2 binomials, with one as the Sum of Square Roots and the other as the Difference of Square Roots.
It can be expressed as:
\((x + y)(x - y)\)
Factoring Difference of Squares
Outlined below are the steps required to factor an expression written as a difference of squares:
- Determine if there is a Greatest Common Factor (GCF) that can be factored out of both terms. Include the GCF for the remainder of the expression
- Rewrite the expression in the form \((x)^2 - (y)^2\). Do this by finding the square root of both terms
- Use the formula \((a + b)(a – b)\) to factor the rest of the expression
Example
Factor the binomial \(4x^2 - 25\)
First, as there's no common factor, we can skip Step 1.
Next, we can rewrite the expression in the form \((x)^2 - (y)^2\):
\(= \sqrt{4x^2} - \sqrt{25}\)
\(= 2x - 5\)
\(= (2x)^2 - (5)^2\)
Then, we can use the formula \((x + y)(x - y)\) to factor the rest of the expression:
\(= (2x + 5)(2x - 5)\)
Therefore, we can determine that \(4x^2 - 25\) expressed as a Difference of Squares is \((2x + 5)(2x - 5)\).
Factor the following binomials:
\(121x^2 - 9y^2\)
Show Answer
First, as there's no common factor, we can skip Step 1.
Next, we can rewrite the expression in the form \((x)^2 - (y)^2\) by taking the square root of each term then raising them to the power \(2\):
\(= \sqrt{121x^2} - \sqrt{9y^2}\)
\(= 11x - 3y\)<
\(= (11x)^2 - (3y)^2\)
Then, we can use the formula \((x + y)(x - y)\) to factor the rest of the expression:
\(= (11x + 3y)(11x - 3y)\)
Therefore, we can determine that \(121x^2 - 9y^2\) factored is \((11x + 3y)(11x - 3y)\).
\(18x^2 - 32\)
Show Answer
First, we can factor out the GCF (in this case, \(2\)):
\(= 2(9x^2 - 16)\)
Next, we can rewrite the expression in the form \((x)^2 - (y)^2\):
\(= 2(\sqrt{9x^2} - \sqrt{16})\)
\(= 2(3x - 4)\)
\(= 2((3x)^2 - (4)^2)\)
Then, we can use the formula \((x + y)(x - y)\) to factor the rest of the expression:
\(=2(3x + 4)(3x - 4)\)
Therfore, we can determine that \(18x^2 - 32\) factored is \(2(3x + 4)(3x - 4)\).
\((5c + 3)^2 - (2c + 1)^2\)
Show Answer
First, as there's no common factor, we can skip Step 1. Likewise, as both terms are already squared, we can skip Step 2.
Next, we can use the formula \((x + y)(x - y)\) to factor and simplify the rest of the expression:
\(= (5c + 3) + (2c + 1))((5c + 3 - (2c + 1))\)
\(= (7c + 4)(3c + 2)\)
Therefore, we can determine that \((5c + 3)^2 - (2c + 1)^2\) factored is \((7c + 4)(3c + 2)\).
Find an algebraic expression for the area of the shaded region in factored form.
Show Answer
First, we can find the area of the shaded area by finding the difference between the area of the whole shape and the area of the white area. This can be expressed as a Difference of Squares:
\(A = (3x + 4)^2 - (x - 2)^2\)
Next, we can expand and simplify the expression as such:
\(A = [(3x + 4) + (x - 2)][(3x + 4) - (x - 2)]\)
\(A = (4x - 2)(3x + 4 - x + 2)\)
\(A = (4x - 2)(2x + 6)\)
Finally, we can remove out the GCF from both binomials to further simplify the expression:
\(A = 2(2x - 1)2(x - 3)\)
\(A = 4(2x - 3)(x - 1)\)
Therefore, we can determine that the area of the shaded region can be expressed as \(4(2x - 3)(x - 1)\).