In algebra class, the teacher always discusses the topic of the sum of cubes and the difference of two cubes side by side. The reason is that they are similar in structure. The key is to “memorize” or remember the patterns involved in the formulas. Here are the steps required for factoring a sum of cubes: Decide if the two terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF.

Do not forget to include the GCF as part of your final answer. Rewrite the original problem as a difference of two perfect cubes. Write What You See”, “Square-Multiply-Square”, “Same, Different, End on a Positive”. Use these three pieces to write the final answer.

Sum Of Cubes, Sum Of Two Cubes & More

The polynomial in the form a³+b³ is called the sum of two cubes because two cubic terms are being added together.

{a^3} + {b^3}
{a^3} + {b^3}

For the “sum” case, the binomial factor on the right side of the equation has a middle sign that is positive. In addition to the “sum” case, the middle sign of the trinomial factor will always be opposite the middle sign of the given problem. Therefore, it is negative.

Sum Of Cubes Formula

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
Sum Of Cubes Formula


Factor Sum Of Cubes

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.

Factor Sum Of Cubes
Factor Sum Of Cubes

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.

Factor x3 – 8

This is equivalent to x3 – 23. With the “minus” sign in the middle, this is a difference of cubes. To do the factoring, I’ll be plugging x and 2 into the difference-of-cubes formula. Doing so, I get:

x3 – 8 = x3 – 23

= (x – 2)(x2 + 2x + 22)

(x – 2)(x2 + 2x + 4)

Factor 27x3 + 1

The first term contains the cube of 3 and the cube of x. But what about the second term?

Before panicking about the lack of an apparent cube, I remember that 1 can be regarded as having been raised to any power I like since 1 to any power is still just 1. In this case, the power I’d like is 3, since this will give me a sum of cubes. This means that the expression they’ve given me can be expressed as:

(3x)3 + 13

So, to factor, I’ll be plugging 3x and 1 into the sum-of-cubes formula. This gives me:

27x3 + 1 = (3x)3 + 13

= (3x + 1)((3x)2 – (3x)(1) + 12)

(3x + 1)(9x2 – 3x + 1)

Factor x3y6 – 64

First, I note that they’ve given me a binomial (a two-term polynomial) and that the power on the x in the first term is 3 so, even if I weren’t working in the “sums and differences of cubes” section of my textbook, I’d be on notice that maybe I should be thinking in terms of those formulas. Looking at the other variable, I note that a power of 6 is the cube of a power of 2, so the other variable in the first term can be expressed in terms of cubing, too; namely, as the cube of the square of y.

The second term is 64,  which I remember is the cube of 4.  So I now know that, with the “minus” in the middle, this is a difference of two cubes; namely, this is:

(xy2)3 – 43

Plugging into the appropriate formula, I get:

x3y6 – 64 = (xy2)3 – 43

= (xy2 – 4)((xy2)2 + (xy2)(4) + 42)

(xy2 – 4)(x2y4 + 4xy2 + 16)


How Do You Factor The Sum Or Difference Of Cubes?

The Distinction Between The Two Formulas Is In The Location Of That One “Minus” Sign: For The Difference Of Cubes, The “Minus” Sign Goes In The Linear Factor, A – B; For The Sum Of Cubes, The “Minus” Sign Goes In The Quadratic Factor, A2 – Ab + B2.

What Is The Identity Of A 3 B 3?

(A + B + C)2 = A2 + B2 + C2 + 2Ab + 2Bc + 2Ca. (A + B)3 = A3 + B3 + 3Ab (A + B) (A – B)3 = A3 – B3 – 3Ab (A – B) A3 + B3 + C3– 3Abc = (A + B + C)(A2 + B2 + C2 – Ab – Bc – Ca)

What Is The Cube Formula?

The Cube Is A Three-dimensional Structure Which Is Formed When Six Identical Squares Bind To Each Other In An Enclosed Form. A Cube Generally Has 6 Faces, 12 Edges, And 8 Vertices. When A Number Is Multiplied By Itself Three Times, Then The Formed Is A Cube Number. For Example, 3 × 3 ×3 = 27 Is A Cube Number.

What Is A Perfect Cube?

A Perfect Cube Is A Result Of Multiplying A Number Three Times By Itself. A · A · A= A³ We Can Also Say That Perfect Cubes Are The Numbers That Have Exact Cube Roots. 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000, 1,331, 1,728, 2,197, 2,744, 3,375

What Is 2 Cubed Mean?

In Math, A Cube Is A Number Multiplied By Itself Three Times. The Cube Of 2 Is 8 (2 X 2 X 2). It Is Also A Three-dimensional Shape Where Each Of The Six Sides Is A Square Or Something Shaped Like A Cube, Such As An Ice Cube Or Meat Cut Into Cubes).