Algebra 2 · Unit 0
\(a^2-b^2=(a-b)(a+b)\) · Watch for sums of squares · Pull out a GCF first when you can
Difference of squares
Both terms are perfect squares, and they are subtracted.
$$a^2-b^2=(a-b)(a+b)$$
$$x^2-25=(x-5)(x+5)$$
A sum is not this pattern
\(x^2+81\) is a sum of squares. It is prime over the reals. Do not write \((x+9)(x-9)\).
$$x^2+81 \text{ is prime}$$
Coefficients can be squares too
Rewrite each term as something squared, then use the same formula.
$$9x^2-36=(3x)^2-6^2$$
$$(3x-6)(3x+6)$$
GCF first on hard problems
If the two terms share a common factor, factor it out. What remains should look Easy or Medium.
$$5x^2-45=5(x^2-9)=5(x-3)(x+3)$$
Factor. \(x^2-49\)
This is a difference of two squares: \(x^2-7^2\).
\(x^2-49=(x-7)(x+7)\)
Answer: \((x-7)(x+7)\)
Factor. \(x^2-16\)
\(x^2-16=x^2-4^2\)
\((x-4)(x+4)\)
Answer: \((x-4)(x+4)\)
Factor. \(x^2-36\)
\(x^2-36=x^2-6^2\)
\((x-6)(x+6)\)
Answer: \((x-6)(x+6)\)
Factor. \(x^2+64\)
This is a sum of squares, \(x^2+8^2\), not a difference.
It is not a difference of two squares problem. Prime over the reals.
Answer: prime
Factor. \(x^2-9\)
\(x^2-9=x^2-3^2\)
\((x-3)(x+3)\)
Answer: \((x-3)(x+3)\)
Factor. \(x^2+4\)
This is a sum of squares, \(x^2+2^2\), not a difference.
It is not a difference of two squares problem. Prime over the reals.
Answer: prime
Like \(9x^2-36\). Both terms are already perfect squares, including the coefficient.
Factor. \(4x^2-9\)
Rewrite as squares: \((2x)^2-3^2\).
\(4x^2-9=(2x-3)(2x+3)\)
Answer: \((2x-3)(2x+3)\)
Factor. \(4x^2-25\)
\(4x^2-25=(2x)^2-5^2\)
\((2x-5)(2x+5)\)
Answer: \((2x-5)(2x+5)\)
Factor. \(9x^2-16\)
\(9x^2-16=(3x)^2-4^2\)
\((3x-4)(3x+4)\)
Answer: \((3x-4)(3x+4)\)
Factor. \(25x^2-4\)
\(25x^2-4=(5x)^2-2^2\)
\((5x-2)(5x+2)\)
Answer: \((5x-2)(5x+2)\)
Factor. \(16x^2-1\)
\(16x^2-1=(4x)^2-1^2\)
\((4x-1)(4x+1)\)
Answer: \((4x-1)(4x+1)\)
Factor out the GCF first. Then the leftover expression is Easy or Medium.
Factor. \(2x^2-50\)
GCF is 2: \(2x^2-50=2(x^2-25)\).
Now it is Easy: \(x^2-25=(x-5)(x+5)\).
\(2(x-5)(x+5)\)
Answer: \(2(x-5)(x+5)\)
Factor. \(3x^2-48\)
GCF is 3: \(3x^2-48=3(x^2-16)\).
\(x^2-16=(x-4)(x+4)\)
Answer: \(3(x-4)(x+4)\)
Factor. \(5x^2-20\)
GCF is 5: \(5x^2-20=5(x^2-4)\).
\(x^2-4=(x-2)(x+2)\)
Answer: \(5(x-2)(x+2)\)
Factor. \(7x^2-63\)
GCF is 7: \(7x^2-63=7(x^2-9)\).
\(x^2-9=(x-3)(x+3)\)
Answer: \(7(x-3)(x+3)\)
Factor. \(18x^2-8\)
GCF is 2: \(18x^2-8=2(9x^2-4)\).
Now it is Medium: \(9x^2-4=(3x)^2-2^2=(3x-2)(3x+2)\).
Answer: \(2(3x-2)(3x+2)\)
Factor. \(50x^2-18\)
GCF is 2: \(50x^2-18=2(25x^2-9)\).
Now it is Medium: \(25x^2-9=(5x)^2-3^2=(5x-3)(5x+3)\).
Answer: \(2(5x-3)(5x+3)\)
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