# 7.5 General strategy for factoring polynomials

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By the end of this section, you will be able to:
• Recognize and use the appropriate method to factor a polynomial completely

Before you get started, take this readiness quiz.

1. Factor ${y}^{2}-2y-24$ .
If you missed this problem, review [link] .
2. Factor $3{t}^{2}+17t+10$ .
If you missed this problem, review [link] .
3. Factor $36{p}^{2}-60p+25$ .
If you missed this problem, review [link] .
4. Factor $5{x}^{2}-80$ .
If you missed this problem, review [link] .

## Recognize and use the appropriate method to factor a polynomial completely

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.

## Factor polynomials.

1. Is there a greatest common factor?
• Factor it out.
2. Is the polynomial a binomial, trinomial, or are there more than three terms?
• If it is a binomial:
Is it a sum?
• Of squares? Sums of squares do not factor.
• Of cubes? Use the sum of cubes pattern.
Is it a difference?
• Of squares? Factor as the product of conjugates.
• Of cubes? Use the difference of cubes pattern.
• If it is a trinomial:
Is it of the form ${x}^{2}+bx+c$ ? Undo FOIL.
Is it of the form $a{x}^{2}+bx+c$ ?
• If $a$ and $c$ are squares, check if it fits the trinomial square pattern.
• Use the trial and error or “ac” method.
• If it has more than three terms:
Use the grouping method.
3. Check.
• Is it factored completely?
• Do the factors multiply back to the original polynomial?

Remember, a polynomial is completely factored if, other than monomials, its factors are prime!

Factor completely: $4{x}^{5}+12{x}^{4}$ .

## Solution

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

Factor completely: $3{a}^{4}+18{a}^{3}$ .

$3{a}^{3}\left(a+6\right)$

Factor completely: $45{b}^{6}+27{b}^{5}$ .

$9{b}^{5}\left(5b+3\right)$

Factor completely: $12{x}^{2}-11x+2$ .

## Solution

 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.

$\phantom{\rule{2.5em}{0ex}}\left(3x-2\right)\left(4x-1\right)$

$\phantom{\rule{2.5em}{0ex}}12{x}^{2}-3x-8x+2$

$\phantom{\rule{2.5em}{0ex}}12{x}^{2}-11x+2\phantom{\rule{0.2em}{0ex}}✓$

Factor completely: $10{a}^{2}-17a+6$ .

$\left(5a-6\right)\left(2a-1\right)$

Factor completely: $8{x}^{2}-18x+9$ .

$\left(2x-3\right)\left(4x-3\right)$

Factor completely: ${g}^{3}+25g$ .

## Solution

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

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: $12{y}^{2}-75$ .

## Solution

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

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