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Completion requirements
Fully factor the following expressions.
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\(3x^2 - 9x\)
GCF = \(3x\)
\(\begin{align}
\frac{{3x^2 - 9x}}{{3x}} &= \frac{{3x^2 }}{{3x}} - \frac{{9x}}{{3x}} = x - 3 \\
3x^2 - 9x &= 3x\left( {x - 3} \right) \\
\end{align}\)
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\(x^2 - x - 30\)
No GCF
Product of \(-30 = -6(5)\)
Sum of \(-1 = -6 + 5\)
\(\begin{align}
x^2 - x - 30 &= x^2 - 6x + 5x - 30 \\
&= x\left( {x - 6} \right) + 5\left( {x - 6} \right) \\
&= \left( {x - 6} \right)\left( {x + 5} \right) \\
\end{align}\)
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\(12x^2 - 5x - 3\)
No GCF
Product of \(12(-3) = -36 = -9(4)\)
Sum of \(-5 = -9 + 4\)
\(\begin{align}
12x^2 - 5x - 3 &= 12x^2 - 9x + 4x - 3 \\
&= 3x\left( {4x - 3} \right) + \left( {4x - 3} \right) \\
&= \left( {4x - 3} \right)\left( {3x + 1} \right) \\
\end{align}\)
Notice the coefficient in front of the second \(4x - 3\) is \(1\).
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\(2n^2 + 9n + 4\)
No GCF
Product of \(2(4) = 8 = 8(1)\)
Sum of \(9 = 8 + 1\)
\(\begin{align}
2n^2 + 9n + 4 &= 2n^2 + 8n + 1n + 4 \\
&= 2n\left( {n + 4} \right) + \left( {n + 4} \right) \\
&= \left( {n + 4} \right)\left( {2n + 1} \right) \\
\end{align}\)