More Factoring Methods

Learning Outcomes

  • Factor polynomials with negative or fractional exponents
  • Factor by substitution

Expressions with fractional or negative exponents can be factored using the same factoring techniques as those with integer exponents. It is important to remember a couple of things first.

  • When you multiply two exponentiated terms with the same base, you can add the exponents: [latex]x^{-1}\cdot{x^{-1}}=x^{-1+(-1)}=x^{-2}[/latex]
  • When you add fractions, you need a common denominator: [latex]\frac{1}{2}+\frac{1}{3}=\frac{3}{3}\cdot\frac{1}{2}+\frac{2}{2}\cdot\frac{1}{3}=\frac{3}{6}+\frac{2}{6}=\frac{5}{6}[/latex]
  • Polynomials have positive integer exponents – if it has a fractional or negative exponent it is an expression.

First, practice finding a GCF that is a negative exponent.


Factor [latex]12y^{-3}-2y^{-2}[/latex].

Now let us factor a trinomial that has negative exponents.


Factor [latex]x^{-2}+5x^{-1}+6[/latex].

In the next example, we will see a difference of squares with negative exponents. We can use the same shortcut as we have before, but be careful with the exponent.


Factor [latex]25x^{-4}-36[/latex].

In the following video, you will see more examples that are similar to the previous three written examples.

 Fractional Exponents

Again, we will first practice finding a GCF that has a fractional exponent.


Factor [latex]x^{\frac{2}{3}}+3x^{\frac{1}{3}}[/latex].

In our next example, we will factor a perfect square trinomial that has fractional exponents.


Factor [latex]25y^{\frac{1}{2}}+70x^{\frac{1}{4}}+49[/latex].

In our next video, you will see more examples of how to factor expressions with fractional exponents.

Factor Using Substitution

We are going to move back to factoring polynomials; our exponents will be positive integers. Sometimes we encounter a polynomial that looks similar to something we know how to factor but is not quite the same. Substitution is a useful tool that can be used to “mask” a term or expression to make algebraic operations easier.

You may recall that substitution can be used to solve systems of linear equations and to check whether a point is a solution to a system of linear equations.

For example, consider the following system of equations:

[latex]\begin{array}{l}x+3y=8\hfill \\ 2x - 9=y\hfill \end{array}[/latex]
To determine whether the ordered pair [latex]\left(5,1\right)[/latex] is a solution to the given system of equations, we can substitute the ordered pair [latex]\left(5,1\right)[/latex] into both equations.

[latex]\begin{array}{ll}\left(5\right)+3\left(1\right)=8\hfill & \hfill \\ \text{ }8=8\hfill & \text{True}\hfill \\ 2\left(5\right)-9=\left(1\right)\hfill & \hfill \\ \text{ }\text{1=1}\hfill & \text{True}\hfill \end{array}[/latex]

We replaced the variable with a number and then performed the algebraic operations specified. In the next example, we will see how we can use a similar technique to factor a fourth degree polynomial.


Factor [latex]x^4+3x^2+2[/latex].

In the following video, we show two more examples of how to use substitution to factor a fourth degree polynomial and an expression with fractional exponents.

Factor Completely

Sometimes you may encounter a polynomial that takes an extra step to factor. In our next example, we will first find the GCF of a trinomial, and after factoring it out, we will be able to factor again so that we end up with a product of a monomial and two binomials.


Factor [latex]6m^2k-3mk-3k[/latex] completely.

In our last example, we show why it is important to factor out a GCF, if there is one, before you begin using the techniques shown in this module.


In this section, we used factoring with special cases and factoring by grouping to factor expressions with negative and fractional exponents. We also returned to factoring polynomials and used the substitution method to factor a [latex]4th[/latex] degree polynomial. The last topic we covered was what it means to factor completely.