In this explainer, we will learn how to factor the sum and the difference of two cubes.
To understand the sum and difference of two cubes, let us first recall a very similar concept: the difference of two squares. For two real numbers and , we have
This factoring of the difference of two squares can be verified by expanding the parentheses on the right-hand side of the equation. We find that
As we can see, this formula works because even though two binomial expressions normally multiply together to make four terms, the and terms in the middle end up canceling out. This identity is useful since it allows us to easily factor quadratic expressions if they are in the form . We might wonder whether a similar kind of technique exists for cubic expressions. In other words, is there a formula that allows us to factor ?
Let us investigate what a factoring of might look like. We might guess that one of the factors is , since it is also a factor of . Supposing that this is the case, we can then find the other factor using long division:
Since the remainder after dividing is zero, this shows that is indeed a factor and that the correct factoring is
One might wonder whether the expression can be factored further since it is a quadratic expression, however, this is actually the most simplified form that it can take (although we will not prove this in this explainer). We note that as and can be any two numbers, this is a formula that applies to any expression that is a difference of two cubes. Specifically, we have the following definition.
Definition: Difference of Two Cubes
For two real numbers and , the expression is called the difference of two cubes. It can be factored as follows:
We can additionally verify this result in the same way that we did for the difference of two squares. If we expand the parentheses on the right-hand side of the equation, we find
Much like how the middle terms cancel out in the difference of two squares, we can see that the same occurs for the difference of cubes.
This result is incredibly useful since it gives us an easy way to factor certain types of cubic equations that would otherwise be tricky to factor. Let us see an example of how the difference of two cubes can be factored using the above identity.
Example 1: Finding an Unknown by Factoring the Difference of Two Cubes
Given that , find an expression for .
This question can be solved in two ways. One way is to expand the parentheses on the right-hand side of the equation and find what value of satisfies both sides. An alternate way is to recognize that the expression on the left is the difference of two cubes, since . This allows us to use the formula for factoring the difference of cubes.
Recall that we have
Since the given equation is , we can see that if we take and , it is of the desired form. This means that must be equal to .
To show how this answer comes about, let us examine what would normally happen if we tried to expand the parentheses. We would have
In order for this expression to be equal to , the terms in the middle must cancel out. These terms have been factored in a way that demonstrates that choosing leads to both terms being equal to zero. If we do this, then both sides of the equation will be the same.
Therefore, we can confirm that satisfies the equation.
In the previous example, we demonstrated how a cubic equation that is the difference of two cubes can be factored using the formula with relative ease. We note, however, that a cubic equation does not need to be in this exact form to be factored. Suppose, for instance, we took in the formula for the factoring of the difference of two cubes. Then, we would have
Using the fact that and , we can simplify this to get
We also note that is in its most simplified form (i.e., it cannot be factored further). This leads to the following definition, which is analogous to the one from before.
Definition: Sum of Two Cubes
For two real numbers and , the expression is called the sum of two cubes. It can be factored as follows:
Let us verify once more that this formula is correct by expanding the parentheses on the right-hand side. We have
Just as for previous formulas, the middle terms end up canceling out each other, leading to an expression with just two terms.
Let us demonstrate how this formula can be used in the following example.
Example 2: Factoring a Sum of Two Cubes
Factorize fully .
Note that although it may not be apparent at first, the given equation is a sum of two cubes. To see this, let us look at the term . We can see this is the product of 8, which is a perfect cube, and , which is a cubic power of . Therefore, if we take the cube root, we find
So, is the cube of . Now, we recall that the sum of cubes can be written as
Substituting and into the above formula, this gives us
As demonstrated in the previous example, we should always be aware that it may not be immediately obvious when a cubic expression is a sum or difference of cubes. Sometimes, it may be necessary to identify common factors in an expression so that the result becomes the sum or difference of two cubes. Let us consider an example where this is the case.
Example 3: Factoring a Difference of Two Cubes
Factorize fully .
Before attempting to fully factor the given expression, let us note that there is a common factor of 2 between the terms. Therefore, it can be factored as follows:
From here, we can see that the expression inside the parentheses is a difference of cubes. This is because each of and is a product of a perfect cube number (i.e., and ) and a cubed variable ( and ). In other words, we have
Recall that the difference of two cubes can be written as
Thus, the full factoring is
Let us continue our investigation of expressions that are not evidently the sum or difference of cubes by considering a polynomial expression with sixth-order terms and seeing how we can combine different formulas to get the solution.
Example 4: Factoring a Difference of Squares That Results in a Product of a Sum and Difference of Cubes
Factor the expression .
Although the given expression involves sixth-order terms and we do not have any formula for dealing with them explicitly, we note that we can apply the laws of exponents to help us. Specifically, the expression can be written as a difference of two squares as follows:
Note that it is also possible to write this as the difference of cubes, but the resulting expression is more difficult to simplify. Recall that the difference of two squares can be written as
Letting and here, this gives us
Now, we have a product of the difference of two cubes and the sum of two cubes. Thus, we can apply the following sum and difference formulas:
Thus, we let and and we obtain the full factoring of the expression:
For our final example, we will consider how the formula for the sum of cubes can be used to solve an algebraic problem.
Example 5: Evaluating an Expression Given the Sum of Two Cubes
If and , what is the value of ?
We begin by noticing that is the sum of two cubes. This is because is 125 times , both of which are cubes. So, if we take its cube root, we find
Recall that we have the following formula for factoring the sum of two cubes:
Here, if we let and , we have
Note that we have been given the value of but not . However, it is possible to express this factor in terms of the expressions we have been given. Suppose we multiply with itself:
This is almost the same as the second factor but with added on. In other words, by subtracting from both sides, we have
Since we have been given the value of , the left-hand side of this equation is now purely in terms of expressions we know the value of. Therefore, we can rewrite as follows:
Let us summarize the key points we have learned in this explainer.
- The difference of two cubes can be written as
- Similarly, the sum of two cubes can be written as
- Using substitutions (e.g., or ), we can use the above formulas to factor various cubic expressions.
- By identifying common factors in cubic expressions, we can in some cases reduce them to sums or differences of cubes.
- We can combine the formula for the sum or difference of cubes with that for the difference of squares to simplify higher-order expressions.