Solving by Factoring

This section covers:

We first learned about factoring methods here in the Solving Quadratics by Factoring and Completing the Square section.

Then we got into a little more complicated factoring here in the Rational Functions and Equations section.

Now that we know how to solve by Finding Roots of Polynomial Functions, 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, factorset each factor to 0 and solve.

Factoring and Solving Polynomial Equations

Here are some examples of factoring and solving polynomial equations; solve over the reals:

Solving by Factoring

Now we can do more advanced solving, since we know how to factor higher degree polynomials.  Note that we use the techniques that we learned in the  Finding Roots of Polynomial Functions section.  

Let’s solve over the real and complex numbers:

Factoring with Exponents

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

Note also that we’ll discuss Exponential Functions 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  \({{x}^{-5}}+{{x}^{-2}}+x\), we’d take out  \({{x}^{-5}}\)  for the GCF to get  \({{x}^{-5}}\left( {{x}^{-5-\,-5}}+{{x}^{-2-\,-5}}+{{x}^{1-\,-5}} \right)={{x}^{-5}}\left( {{x}^{0}}+{{x}^{3}}+{{x}^{6}} \right)={{x}^{-5}}\left( 1+{{x}^{3}}+{{x}^{6}} \right)\).  (Remember that “ – – ” is the same as “+ +” or “+”).   Multiply back to make sure you’ve done it correctly.
  • For fractional coefficients, find the common denominator, and take out the fraction that goes into all the other fractions.  So for the fraction you take out, the denominator is the least common denominator of all the fractions, and the numerator is the Greatest Common Factor (GCF) of the numerators.  For example, for the expression  \(\frac{3}{4}{{x}^{2}}-\frac{2}{4}x+\frac{16}{4}=\frac{1}{4}\left( 3{{x}^{2}}-2x+16 \right)\)  (since nothing except for 1 goes into 3 and 2 and 16).  Multiply back to make sure you’ve done it 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 \(\left( u-2 \right)\left( u+1 \right)\).    We can then translate 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 foil back to make sure you’ve done it correctly.  We call this method u-substitution or simply u-sub.

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

Factoring Exponential Expressions

Solving Exponential Equations

After factoring, you may be asked to solve the exponential equation.  Here are some examples, some using u-substitution.   Note that the second problem uses the concept of Changing the Base of an Exponent, which we’ll talk about here.

Factoring and Solving with Exponents

Learn these rules, and practice, practice, practice!

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the Mathway site, where you can register for the full version (steps included) of the software.  You can even get math worksheets.

You can also go to the Mathway site here, where you can register, or just use the software for free without the detailed solutions.  There is even a Mathway App for your mobile device.  Enjoy!

On to Exponential Functions – you are ready!