Revisiting Factoring Quadratics | Factoring and Solving with Exponents |

Factoring Sum and Difference of Cubes | More Practice |

Factoring and Solving with Polynomials |

**Factoring** is extremely important in math; we first learned factoring here** in the Solving Quadratics by Factoring and Completing the Square** section.

## Revisiting Factoring Quadratics

Earlier, we learned how to factor, or “un**FOIL**” a trinomial into two binomials. (Remember that FOIL stands for First-Outer-Inner-Last when multiplying two binomials together):

Sometimes we have to factor out common factors, or **GCF**s. We always want to do this first:

And don’t forget “grouping” when we have four terms (but it doesn’t always work – we found other ways to solve in **Graphing and Finding Roots of Polynomial Functions** later). Again, we are working with **GCF**‘s to do this:

You can factor a **difference of squares**, but not a sum of squares: $ 9{{x}^{2}}-25=\left( {3x-5} \right)\left( {3x+5} \right)$. However, $ 9{{x}^{2}}+25$ is **prime**, or** irreducible**; you can’t factor with reals, but would be $ \left( {3x+5i} \right)\left( {3x-5i} \right)$ using imaginary numbers)

## Factoring with Sum and Difference of Cubes

For **cubic binomials** (sum or difference of cubes), we can factor:

Here are some examples of **factoring sums and differences of cubes**:

## Factoring and Solving with Polynomials

Again, we first learned about factoring methods here in the **Solving Quadratics by Factoring and Completing the Square** section. Now that we know how to by **Graphing and Finding Roots of Polynomial Functions section** we can do fancier factoring, and thus find more roots. Remember that when we want to find **solutions** or **roots**, we **set the equation to 0**, **factor**, **set each factor to 0** and **solve**. Here are some examples of factoring and solving **polynomial equations**; solve over the **reals**:

Here’s even more advanced solving, using techniques we will learn here in the **Graphing and Finding Roots of Polynomial Functions **section**. **Solve over the **real** and **complex** numbers:

## Factoring and Solving with Exponents

Factoring and Solving with **Exponential Functions** can be a bit trickier. Note that we learned about the **properties of exponents** here in the **Exponents and Radicals in Algebra** section, and did some solving with exponents **here**.

In your Pre-Calculus and Calculus classes, you may see **algebraic exponential expressions** that need factoring and possibly solving, either by taking out a Greatest Common Factor (**GCF**) or by “unFOILing”. These really aren’t that bad, if you remember a few hints:

- To take out a
**GFC**with exponents, take out the factor with the**smallest exponent**, whether it’s positive or negative. (Remember that “larger” negative numbers are actually smaller). This is even true for**fractional exponents**. Then, to get what’s left in the parentheses after you take out the**GCF**, subtract the exponents from the one you took out. For example, for the expression $ {{x}^{-5}}+{{x}^{-2}}+x$, take out $ {{x}^{-5}}$ for the**GCF**to get $ \displaystyle \begin{array}{l}{{x}^{{-5}}}\left( {{{x}^{{-5-\,-5}}}+{{x}^{{-2-\,-5}}}+{{x}^{{1-\,-5}}}} \right)\\={{x}^{{-5}}}\left( {{{x}^{0}}+{{x}^{3}}+{{x}^{6}}} \right)\\={{x}^{{-5}}}\left( {1+{{x}^{3}}+{{x}^{6}}} \right)\end{array}$. (Remember that “ – – ” is the same as “+ +” or “+”). Multiply back to make sure you’ve factored correctly! - For
**fractional coefficients**, find the common denominator, and take out the fraction that goes into all the other fractions. For the fraction you take out, the denominator is the least common denominator (**LCD**) of all the fractions, and the numerator is the Greatest Common Factor (**GCF**) of the numerators. For example, $ \displaystyle \begin{array}{l}\displaystyle \frac{3}{4}{{x}^{2}}\displaystyle -\frac{1}{2}x+4=\displaystyle \frac{3}{4}{{x}^{2}}\displaystyle -\frac{2}{4}x+\displaystyle \frac{{16}}{4}\\=\displaystyle \frac{1}{4}\left( {3{{x}^{2}}-2x+16} \right)\end{array}$ (since nothing except for**1**goes into**3**and**2**and**16**). Multiply back to make sure you’ve factored correctly! - For a
**trinomial with a constant**, if the largest exponent is twice that of the middle exponent, then use substitution like $ u$, for the middle exponent, “unFOIL”, and then put the “real” expression back in. For example, for $ {{x}^{\frac{2}{3}}}-{{x}^{\frac{1}{3}}}-2$ , let $ u={{x}^{\frac{1}{3}}}$ , and we have $ {{u}^{2}}-u-2$, which factors to $ \left( u-2 \right)\left( u+1 \right)$. Then substitute back to $ \left( {{x}^{\frac{1}{3}}}-2 \right)\left( {{x}^{\frac{1}{3}}}+1 \right)$ and solve from there (set each to**0**and solve). Always multiply back to make sure you’ve factored correctly. We call this methodor simply*u*-substitution.*u*-sub

Let’s do some factoring. Learning to factor these will actually help you a lot when you get to **Calculus:**

After factoring, you may be asked to solve the exponential equation. Here are some examples, some using ** u-substitution. **Note that the third problem uses

**log solving**from here in the

**Logarithmic Functions**section.

**Learn these rules, and practice, practice, practice!**

**For Practice**: Use the **Mathway** widget below to try a **Factoring **problem. Click on **Submit** (the blue arrow to the right of the problem) and click on **Factor** to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on **Tap to view steps**, or **Click Here**, you can register at **Mathway** for a **free trial**, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to **Conics ** – you are ready!