Recognizing and Using the Appropriate Method to Factor a Polynomial Completely

Factor completely: \(4{x}^{5}+12{x}^{4}\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}4{x}^{4}.\hfill & & & \hfill 4{x}^{5}+12{x}^{4}\hfill \\ & & & \text{Factor out the GCF.}\hfill & & & \hfill 4{x}^{4}(x+3)\hfill \\ \text{In the parentheses, is it a binomial, a}\hfill & & & & & & \\ \text{trinomial, or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Of squares? Of cubes?}\hfill & & & & & & \text{No.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & & & & \text{Yes.}\hfill \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}(x+3)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{4}·x+4{x}^{4}·3\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}4{x}^{5}+12{x}^{4}\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}\)

Factor completely: \(12{x}^{2}-11x+2\).

Factor completely: \({g}^{3}+25g\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes,}\phantom{\rule{0.2em}{0ex}}g.\hfill & & & \hfill {g}^{3}+25g\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill g({g}^{2}+25)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum ? Of squares?}\hfill & & & \text{Yes.}\hfill & & & \text{Sums of squares are prime.}\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}g({g}^{2}+25)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}{g}^{3}+25g\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}\)

Factor completely: \(12{y}^{2}-75\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 3.}\hfill & & & \hfill 12{y}^{2}-75\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 3(4{y}^{2}-25)\hfill \\ \text{In the parentheses, is it a binomial, trinomial},\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & \\ \text{Is it a sum?}\hfill & & & \text{No.}\hfill & & & \\ \text{Is it a difference? Of squares or cubes?}\hfill & & & \text{Yes, squares.}\hfill & & & \hfill 3({(2y)}^{2}-{(5)}^{2})\hfill \\ \text{Write as a product of conjugates.}\hfill & & & & & & \hfill 3(2y-5)(2y+5)\hfill \\ \text{Check.}\hfill & & & & & & \\ \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Neither binomial is a difference of}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{squares.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}3(2y-5)(2y+5)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}3(4{y}^{2}-25)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}12{y}^{2}-75\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}\)

Factor completely: \(4{a}^{2}-12ab+9{b}^{2}\).

Solution

Is there a GCF?	No.
Is it a binomial, trinomial, or are there more terms?
Trinomial with \(a\ne 1\). But the first term is a   perfect square.
Is the last term a perfect square?	Yes.
Does it fit the pattern, \({a}^{2}-2ab+{b}^{2}?\)	Yes.
Write it as a square.
Check your answer.
Is the expression factored completely?
Yes.
The binomial is not a difference of squares.
Multiply.
\({(2a-3b)}^{2}\)
\({(2a)}^{2}-2\cdot 2a\cdot 3b+{(3b)}^{2}\)
\(4{a}^{2}-12ab+9{b}^{2}✓\)

Factor completely: \(6{y}^{2}-18y-60\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 6.}\hfill & & & \hfill 6{y}^{2}-18y-60\hfill \\ \text{Factor out the GCF.}\hfill & & & \text{Trinomial with leading coefficient 1.}\hfill & & & \hfill 6({y}^{2}-3y-10)\hfill \\ \begin{array}{c}\text{In the parentheses, is it a binomial, trinomial,}\hfill \\ \text{or are there more terms?}\hfill \end{array}\hfill & & & & & & \\ \text{“Undo” FOIL.}\hfill & & & \hfill 6(y\phantom{\rule{1em}{0ex}})(y\phantom{\rule{1em}{0ex}})\hfill & & & \hfill 6(y+2)(y-5)\hfill \\ \text{Check your answer.}\hfill & & & & & & \\ \text{Is the expression factored completely?}\hfill & & & & & & \hfill \text{Yes.}\hfill \\ \text{Neither binomial is a difference of squares.}\hfill & & & & & & \\ \text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6(y+2)(y-5)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6({y}^{2}-5y+2y-10)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6({y}^{2}-3y-10)\hfill & & & & & & \\ \phantom{\rule{2.5em}{0ex}}6{y}^{2}-18y-60\phantom{\rule{0.2em}{0ex}}✓\hfill & & & \end{array}\)

Factor completely: \(24{x}^{3}+81\).

Solution

Is there a GCF?	Yes, 3.	\(24{x}^{3}+81\)
Factor it out.		\(3(8{x}^{3}+27)\)
In the parentheses, is it a binomial, trinomial, or are there more than three terms?	Binomial.
Is it a sum or difference?	Sum.
Of squares or cubes?	Sum of cubes.
Write it using the sum of cubes pattern.
Is the expression factored completely?	Yes.	\(3(2x+3)(4{x}^{2}-6x+9)\)
Check by multiplying.		We leave the check to you.

Factor completely: \(2{x}^{4}-32\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 2.}\hfill & & & \hfill 2{x}^{4}-32\hfill \\ \text{Factor it out.}\hfill & & & & & & \hfill 2({x}^{4}-16)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & & & & \\ \text{or are there more than three terms?}\hfill & & & \text{Binomial.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Is it a sum or difference?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{Of squares or cubes?}\hfill & & & \text{Difference of squares.}\hfill & & & \hfill 2({({x}^{2})}^{2}-{(4)}^{2})\hfill \\ \text{Write it as a product of conjugates.}\hfill & & & & & & \hfill 2({x}^{2}-4)({x}^{2}+4)\hfill \\ \text{The first binomial is again a difference of squares.}\hfill & & & & & & \hfill 2({(x)}^{2}-{(2)}^{2})({x}^{2}+4)\hfill \\ \text{Write it as a product of conjugates.}\hfill & & & & & & \hfill 2(x-2)(x+2)({x}^{2}+4)\hfill \\ \text{Is the expression factored completely?}\hfill & & & \text{Yes.}\hfill & & & \\ \phantom{\rule{1em}{0ex}}\text{None of these binomials is a difference of squares.}\hfill & & & & & & \\ \text{Check your answer.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \\ \phantom{\rule{3em}{0ex}}\begin{array}{c}2(x-2)(x+2)({x}^{2}+4)\hfill \\ 2({x}^{2}-4)({x}^{2}+4)\hfill \\ 2({x}^{4}-16)\hfill \\ 2{x}^{4}-32✓\hfill \end{array}\hfill & & & & & & \end{array}\)

Factor completely: \(3{x}^{2}+6bx-3ax-6ab\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 3.}\hfill & & & \hfill 3{x}^{2}+6bx-3ax-6ab\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 3({x}^{2}+2bx-ax-2ab)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & \text{More than 3}\hfill & & & \\ \text{or are there more terms?}\hfill & & & \text{terms.}\hfill & & & \\ \text{Use grouping.}\hfill & & & & & & \hfill 3[x(x+2b)-a(x+2b)]\hfill \\ & & & & & & \hfill 3(x+2b)(x-a)\hfill \\ \text{Check your answer.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Is the expression factored completely? Yes.}\hfill & & & & & & \\ \phantom{\rule{1em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}3(x+2b)(x-a)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}3({x}^{2}-ax+2bx-2ab)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}3{x}^{2}-3ax+6bx-6ab\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}\)

Factor completely: \(10{x}^{2}-34x-24\).

Solution

\(\begin{array}{ccccccc}\text{Is there a GCF?}\hfill & & & \text{Yes, 2.}\hfill & & & \hfill 10{x}^{2}-34x-24\hfill \\ \text{Factor out the GCF.}\hfill & & & & & & \hfill 2(5{x}^{2}-17x-12)\hfill \\ \text{In the parentheses, is it a binomial, trinomial,}\hfill & & & \hfill \text{Trinomial with}\hfill & & & \\ \text{or are there more than three terms?}\hfill & & & a\ne 1.\hfill & & & \\ \text{Use trial and error or the “ac” method.}\hfill & & & & & & \hfill 2\underset{}{(5{x}^{2}}-17x\underset{}{-12)}\hfill \\ & & & & & & \hfill 2(5x+3)(x-4)\hfill \\ \begin{array}{c}\text{Check your answer. Is the expression factored}\hfill \\ \text{completely? Yes.}\hfill \end{array}\hfill & & & & & & \\ \phantom{\rule{2em}{0ex}}\text{Multiply.}\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}2(5x+3)(x-4)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}2(5{x}^{2}-20x+3x-12)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}2(5{x}^{2}-17x-12)\hfill & & & & & & \\ \phantom{\rule{3em}{0ex}}10{x}^{2}-34x-24\phantom{\rule{0.2em}{0ex}}✓\hfill & & & & & & \end{array}\)