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Before you get started, take this readiness quiz.
You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. [link] outlines a strategy you should use when factoring polynomials.
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!
Factor completely: $4{x}^{5}+12{x}^{4}$ .
$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}4{x}^{4}.\hfill & & & \hfill 4{x}^{5}+12{x}^{4}\hfill \\ & & & \text{Factor out the GCF.}\hfill & & & \hfill 4{x}^{4}\left(x+3\right)\hfill \\ \text{In the parentheses, is it a binomial, a}\hfill & & & & & & \\ \text{trinomial, or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Of squares? Of cubes?}\hfill & & & & & & \text{No.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \\ \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}\left(x+3\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}\xb7x+4{x}^{4}\xb73\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{5}+12{x}^{4}\phantom{\rule{0.2em}{0ex}}\u2713\hfill & & & & & & \end{array}$
Factor completely: $3{a}^{4}+18{a}^{3}$ .
$3{a}^{3}\left(a+6\right)$
Factor completely: $45{b}^{6}+27{b}^{5}$ .
$9{b}^{5}\left(5b+3\right)$
Factor completely: $12{x}^{2}11x+2$ .
Is there a GCF?  No.  
Is it a binomial, trinomial, or are
there more than three terms? 
Trinomial.  
Are a and c perfect squares?  No,
a = 12,
not a perfect square. 

Use trial and error or the “ac” method.
We will use trial and error here. 
Check.
$\phantom{\rule{2.5em}{0ex}}\left(3x2\right)\left(4x1\right)$
$\phantom{\rule{2.5em}{0ex}}12{x}^{2}3x8x+2$
$\phantom{\rule{2.5em}{0ex}}12{x}^{2}11x+2\phantom{\rule{0.2em}{0ex}}\u2713$
Factor completely: $10{a}^{2}17a+6$ .
$\left(5a6\right)\left(2a1\right)$
Factor completely: $8{x}^{2}18x+9$ .
$\left(2x3\right)\left(4x3\right)$
Factor completely: ${g}^{3}+25g$ .
$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}g.\hfill & & & \hfill {g}^{3}+25g\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill g\left({g}^{2}+25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum ? Of squares?}\hfill & & & \text{Yes.}\hfill & & & \text{Sums of squares are prime.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}g\left({g}^{2}+25\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}{g}^{3}+25g\phantom{\rule{0.2em}{0ex}}\u2713\hfill & & & & & & \end{array}$
Factor completely: ${x}^{3}+36x$ .
$x\left({x}^{2}+36\right)$
Factor completely: $27{y}^{2}+48$ .
$3\left(9{y}^{2}+16\right)$
Factor completely: $12{y}^{2}75$ .
$\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 3.}\hfill & & & \hfill 12{y}^{2}75\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 3\left(4{y}^{2}25\right)\hfill \\ \text{In the parentheses, is it a binomial, trinomial},\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & \\ \text{Is it a sum?}\hfill & & & \text{No.}\hfill & & & \\ \text{Is it a difference? Of squares or cubes?}\hfill & & & \text{Yes, squares.}\hfill & & & \hfill 3\left({\left(2y\right)}^{2}{\left(5\right)}^{2}\right)\hfill \\ \text{Write as a product of conjugates.}\hfill & & & & & & \hfill 3\left(2y5\right)\left(2y+5\right)\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Neither binomial is a difference of}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{squares.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}3\left(2y5\right)\left(2y+5\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}3\left(4{y}^{2}25\right)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}12{y}^{2}75\phantom{\rule{0.2em}{0ex}}\u2713\hfill & & & & & & \end{array}$
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