Factoring Solutions

• There are many ways to factor an expression. Consider this order for factoring polynomial expressions:


1. 1. Find the greatest common monomial factor    GCF    (other than 1).

   2. Factor by    grouping    (when given 4 terms).

   3. Analyze    sign    patterns.

   4. Factor    special    products.

   5. Factor    completely    using your preferred method.


• Steps 1 and 2 both require determining and    factoring out    the GCF.

• Step 3 refers to these patterns for factored trinomial expressions:

Example	$$\begin{gathered} x^2+7x+10 \\ \left(x+2\right)\left(x+5\right) \end{gathered}$$	$$\begin{gathered} x^2–7x+10 \\ \left(x–2\right)\left(x–5\right) \end{gathered}$$	$$\begin{gathered} x^2–3x–10 \\ \left(x+2\right)\left(x–5\right) \end{gathered}$$	$$\begin{gathered} x^2+3x–10 \\ \left(x–2\right)\left(x+5\right) \end{gathered}$$
Sign Pattern	All terms positive    two (+)	–, + pattern    two (–)	End term (–)    (+) and (–)	End term (–)    (+) and (–)

• Step 4 reminds you to look for:

• Step 5 will be reviewed in Review Lesson 7.

Example 1

Factor by grouping.

$5x^2–20x–6mx+24m$

Implement

$$\begin{gathered} 5x^2–20x+–6mx+24m \\ \left(5x^2–20x\right)+\left(–6mx+24m\right) \end{gathered}$$

$$\begin{gathered} 5x\left(x–4\right)–6m\left(x–4\right) \\ \left(x–4\right)\left(5x–6m\right) \end{gathered}$$

 

Example 2

Factor.

$81x^2–16$

Example 3

Factor.

$25x^2–60x+36$