DIFFERENCE OF CUBES: FORMULAS, EQUATIONS, EXAMPLES, EXERCISES - MATHS - 2025

The difference of cubes is a binomial algebraic expression of the form a ³ - b ³, where the terms a and b can be real numbers or algebraic expressions of various types. An example of a difference of cubes is: 8 - x ³, since 8 can be written as 2 ³.

Geometrically we can think of a large cube, with side a, from which the small cube with side b is subtracted, as illustrated in figure 1:

Figure 1. A difference of cubes. Source: F. Zapata.

The volume of the resulting figure is precisely a difference of cubes:

V = a ³ - b ³

To find an alternative expression, it is observed that this figure can be decomposed into three prisms, as shown below:

Figure 2. The difference of cubes (left of the equality) is equal to the sum of the partial volumes (right). Source: F. Zapata.

A prism has a volume given by the product of its three dimensions: width x height x depth. In this way, the resulting volume is:

V = a ³ - b ³ = a ².b + b ³ + ab ²

Factor b is common to the right. Furthermore, in the figure shown above, it is particularly true that:

b = (a / 2) ⇒ a = b + b

Therefore it can be said that: b = a - b. Thus:

This way of expressing the difference of cubes will prove to be very useful in many applications and it would have been obtained in the same way, even if the side of the missing cube in the corner was different from b = a / 2.

Note that the second parentheses closely resembles the notable product of the square of the sum, but the cross term is not multiplied by 2. The reader can expand the right side to verify that a ³ - b ³ is indeed obtained.

Examples

There are several differences of cubes:

1 - m ⁶

a ⁶ b ³ - 8z ¹² and ⁶

(1/125).x ⁶ - 27.y ⁹

Let's analice each one of them. In the first example, the 1 can be written as 1 = 1 ³ and the term m ⁶ becomes: (m ²) ³. Both terms are perfect cubes, therefore their difference is:

1 - m ⁶ = 1 ³ - (m ²) ³

In the second example the terms are rewritten:

a ⁶ b ³ = (a ² b) ³

8z ¹² y ⁶ = 2 ³ (z ⁴) ³ (y ²) ³ = (2z ⁴ y ²) ³

The difference of these cubes is: (a ² b) ³ - (2z ⁴ y ²) ³.

Finally, the fraction (1/125) is (1/5 ³), x ⁶ = (x ²) ³, 27 = 3 ^3, and y ⁹ = (y ³) ³. Substituting all this in the original expression, you get:

(1/125).x ⁶ - 27y ⁹ = ³ - (3y ³) ³

Factoring a difference of cubes

Factoring the difference of cubes simplifies many algebraic operations. To do this, just use the formula deduced above:

Figure 3. Factorization of the difference of cubes and expression of a remarkable quotient. Source: F. Zapata.

Now, the procedure to apply this formula consists of three steps:

- In the first place the cube root of each of the terms of the difference is obtained.

- Then the binomial and the trinomial that appear on the right side of the formula are constructed.

- Finally, the binomial and the trinomial are replaced to obtain the final factorization.

Let's illustrate the use of these steps with each of the cube difference examples proposed above and thus obtain its factored equivalent.

Example 1

Factor the expression 1 - m ⁶ following the steps described. We start by rewriting the expression as 1 - m ⁶ = 1 ³ - (m ²) ³ to extract the respective cube roots of each term:

Next, the binomial and the trinomial are constructed:

a = 1

b = m ²

So:

a - b = 1 - m ²

(a ² + ab + b ²) = 1 ² + 1.m ² + (m ²) ² = 1 + m ² + m ⁴

Finally, it is substituted in the formula a ³ - b ³ = (ab) (a ² + ab + b ²):

1 - m ⁶ = (1 - m ²) (1 + m ² + m ⁴)

Example 2

Factorize:

a ⁶ b ³ -8z ¹² y ⁶ = (a ² b) ³ - (2z ⁴ y ²) ³

Since these are perfect cubes, the cube roots are immediate: a ² b and 2z ⁴ and ², hence it follows that:

- Binomial: a ² b - 2z ⁴ and ²

- Trinomial: (a ² b) ² + a ² b. 2z ⁴ y ² + (a ² b + 2z ⁴ y ²) ²

And now the desired factorization is constructed:

a ⁶ b ³ -8z ¹² y ⁶ = (a ² b - 2z ⁴ y ²). =

= (a ² b - 2z ⁴ y ²).

In principle, the factoring is ready, but it is often necessary to simplify each term. Then the remarkable product -square of a sum- that appears at the end is developed and then like terms are added. Remembering that the square of a sum is:

The notable product on the right is developed like this:

(a ² b + 2z ⁴ and ²) ² = a ⁴ b ² + 4a ² b.z ⁴ and ² + 4z ⁸ and ⁴

Substituting the expansion obtained in the factorization of the difference of cubes:

a ⁶ b ³ -8z ¹² y ⁶ = (a ² b - 2z ⁴ y ²). =

Finally, grouping like terms and factoring the numerical coefficients, which are all even, we obtain:

(a ² b - 2z ⁴ y ²). = 2 (a ² b - 2z ⁴ y ²).

Example 3

Factoring (1/125) x ⁶ - 27y ⁹ is much easier than the previous case. First the equivalents of a and of b are identified:

a = (1/5) x ²

b = 3y ³

Then they are directly substituted in the formula:

(1/125).x ⁶ - 27y ⁹ =.

Exercise resolved

The difference of cubes has, as we have said, a variety of applications in Algebra. Let's see some:

Exercise 1

Solve the following equations:

a) x ⁵ - 125 x ² = 0

b) 64 - 729 x ³ = 0

Solution to

First the equation is factored in this way:

x ² (x ³ - 125) = 0

Since 125 is a perfect cube, the parentheses are written as a difference of cubes:

x ². (x ³ - 5 ³) = 0

The first solution is x = 0, but we find more if we make x ³ - 5 ³ = 0, then:

x ³ = 5 ³ → x = 5

Solution b

The left side of the equation is rewritten as 64 - 729 x ³ = 4 ³ - (9x) ³. Thus:

4 ³ - (9x) ³ = 0

Since the exponent is the same:

9x = 4 → x = 9/4

Exercise 2

Factor the expression:

(x + y) ³ - (x - y) ³

Solution

This expression is a difference of cubes, if in the factoring formula we note that:

a = x + y

b = x- y

Then the binomial is constructed first:

a - b = x + y - (x- y) = 2y

And now the trinomial:

a ² + ab + b ² = (x + y) ² + (x + y) (xy) + (xy) ²

Notable products are developed:

Next you have to substitute and reduce like terms:

a ² + ab + b ² = x ² + 2xy + y ² + x ² - y ² + x ² - 2xy + y ² = 3x ² + y ²

Factoring results in:

(x + y) ³ - (x - y) ³ = 2y. (3x ² + y ²)

References

1. Baldor, A. 1974. Algebra. Editorial Cultural Venezolana SA
2. CK-12 Foundation. Sum and difference of cubes. Recovered from: ck12.org.
3. Khan Academy. Factoring of differences of cubes. Recovered from: es.khanacademy.org.
4. Math is Fun Advanced. Difference of two cubes. Recovered from: mathsisfun.com
5. UNAM. Factoring a difference of cubes. Recovered from: dcb.fi-c.unam.mx.