Read: More Factoring Methods
Learning Objectives
- Factor expressions with negative exponents
- Factor expressions with fractional exponents
- Factor by substitution
- Factor completely
- When you multiply two exponentiated terms with the same base, you can add the exponents:
- When you add fractions, you need a common denominator:
- Polynomials have positive integer exponents - if it has a fractional or negative exponent it is an expression.
Example
FactorAnswer: If the exponents in this expression were positive we could determine that the GCF is , but since we have negative exponents, we will need to use . Therefore We can check that we are correct by multiplying:
Answer
Example
Factor .Answer: If the exponents on this trinomial were positive, we could factor this as . Note that the exponent on the x's in the factored form is , in other words . Also note that , therefore if we factor this trinomial as , we will get the correct result if we check by multiplying.
Answer
Example
FactorAnswer: Recall that a difference of squares factors in this way: , and the first thing we did was identify a and b to see whether we could factor this as a difference of squares. Given , we can define because Therefore the factored form is:
Answer
Fractional Exponents
Again, we will first practice finding a GCF that has a fractional exponent.Example
FactorAnswer: First, look for the term with the lowest value exponent. In this case, it is . Recall that when you multiply terms with exponents, you add the exponents. To get you would need to add to , so we will need a term whose exponent is . , therefore:
Answer
Example
FactorAnswer: Recall that a perfect square trinomial of the form factors as The first step in factoring a perfect square trinomial was to identify a and b. To find a, we ask: , and recall that , therefore we are looking for an exponent for x that when multiplied by , will give . You can also think about the fact that the middle term is defined as so a will probably have an exponent of , therefore a choice for a may be We can check that this is right by squaring a: Now we can check whether Our terms work out, so we can use the shortcut to factor:
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 isn't 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: To determine whether the ordered pair is a solution to the given system of equations.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.
Example
FactorAnswer: This looks a lot like a trinomial that we know how to factor except for the exponents. If we substitute , and recognize that we may be able to factor this beast! Everywhere there is an we will replace it with a u, then factor. We aren't quite done yet, we want to factor the original polynomial which had x as it's variable, so we need to replace now that we are done factoring.
Answer
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.Example
Factor completely .Answer: Whenever you factor, first try the easy route and ask yourself if there is a GCF. In this case, there is one, and it is . Factor from the trinomial: We are left with a trinomial that can be factored using your choice of factoring method. We will create a table to find the factors of that sum to
Factors of | Sum of Factors |
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Regroup and find the GCF of each group:
Now factor from each term:
Don't forget the original GCF that we factored out! Our final factored form is: