Learn how to determine if an expression can be factored as a sum of cubes and also how to use the sum of cubes formula to factor these types of expressions.
Previously you have probably come across factoring problems where an expression had two terms, such as x^2 – 4 or 25x^4 – 16.
If both terms were squares and had opposite signs (i.e. one term was positive and one term was negative), then you could factor it as a difference of squares using the difference of squares formula a^2 – b^2 = (a + b) (a – b).
But what should you do when you come across a two-term expression where the terms each have the same sign? Or if the terms are not squares, but are, instead, cubes?
If both terms are cubes, then it may be possible to factor the expression as either a difference of cubes or a sum of cubes, depending on the signs of the terms. An expression with opposite signs (ex. x^3 – 8) could be a difference of cubes, which is covered in a separate lesson. An expression where both terms have the same sign (ex. y^3 + 1), either both positive or both negative, could be factored as a sum of cubes, which is the focus of this lesson.
A sum of cubes is a two-term expression where both terms are cubes and each term has the same sign. It is factored according to the following formula.
sum of cubes formula
Next, we’ll see how you can determine if an expression can be factored as a sum of cubes.
Is an Expression a Sum of Cubes?
An expression must meet two criteria in order to be factored as a sum of cubes. First, each term must be a cube. In other words, each term must be the result of multiplying the same expression by itself three times. Below are some examples:
x^3 is a cube because it is a result of x multiplied by itself three times (x*x*x).
27 is a cube because it is the result of 3 multiplied by itself three times (3*3*3).
Additionally, you may find a cube that contains both numbers and variables. For example, 64z^9 is a cube because there is an expression (4z^3) that, when multiplied by itself three times (4z^3)(4z^3)(4z^3), will equal 64z^9. It’s important to note that every part of each term must be a cube; 7x^6 and 8y^2 are not cubes because 7 is not a cube (although x^6 is) and y^2 is not a cube (although 8 is).
Second, each term must have the same sign, usually both positive. Note that if both signs are negative, you can factor a -1 out of both terms to make them each positive. If both terms have opposite signs, then you may want to try and factor the expression as a difference of squares or a difference of cubes. Now let’s see how you use the sum of cubes formula to factor a problem.
How to Factor a Sum of Cubes
For a sum of cubes, you’ll use the formula mentioned above–
sum of cubes formula
Note that a and b represent the individual expressions that are cubed. They could each be a variable (x), a number (3) or some combination of both (4y^2). First, you must determine what a and b are. Essentially you’re asking, what do I cube to get the first term and what do I cube to make the second term? After you’ve done that, you will plug in the expressions you found for a and b into the formula and simplify them to finish the factoring. Let’s see some examples.
Can the following expressions be factored as a sum of cubes? If yes, factor.
Example 1: y^5 + 27. No, this expression cannot be factored as a sum of cubes because the first term (y^5) is not a cube. In other words, there is nothing that can be multiplied by itself three times to equal y^5. Therefore, the expression cannot be factored.
Example 2: x^3 + 64. Yes, this expression can be factored as a sum of cubes since both terms have the same sign (+) and each expression is a cube. x can be cubed to give x^3 and 4 can be cubed to make 64. Thus, plugging a=x and b=3 into the sum of cubes formula gives (x + 3) (x^2 – (x)(4) + 4^2). Simplifying (x)(4) to 4x and 4^2 to 16 gives us the final answer of (x + 3) (x^2 – 4x + 16).
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