Difference of Cubes Formula: Solved Examples

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Difference of Cubes is a polynomial expression where the difference of two perfect cubes is expressed as a product of a binomial and a trinomial. Difference of two cubes can be factored into a product of a binomial. This can be expressed as x³+y³=(x+y)(x²−xy+y²) and x³−y³=(x−y)(x²+xy+y²). Difference of cubes formula can be used to factorize binomials of cubes. It is also known as a³ - b³ formula.

Table of Content

Difference of Cubes
Geometrical Representation of Difference of Cubes
Things to Remember
Sample Questions

Key Terms: Cubes, polynomial, binomial, trinomial, difference, factor, product

Difference of Cubes

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The cubic polynomial expression showing difference of cubes will have

Two terms
The terms will be separated by a negative sign
Each term will be a perfect cube
The difference of the cubes will be factored into a binomial x trinomial
The binomial will be cube root of first term minus the cube root of the second term
The trinomial will be the square of the first term plus the product of the two terms plus the square of the second term.

Difference of Cubes

Difference of Cubes

Let’s say the two terms are a and b, the difference of cubes will be written as

a³ – b³

When we factor the above binomial expression, we will get two factors. The first factor will be a binomial and the second factor will be a trinomial and the equation will be as follows:

a³ – b³ = (a – b) (a² + ab + b²)

So, the difference of two cubes is equal to the difference of their cube roots i.e. (a - b) times a trinomial (a² + ab + b²), which contains the squares of the cube roots i.e. a² and b²and the opposite of the product of the cube roots i.e. (+ab).

Example of Difference of Cubes

Example of Difference of Cubes

To remember the signs of the factorization use the mnemonic "SOAP",

S - Same sign" as in the middle of the original expression

O- Opposite sign

AP - Always Positive

a³ – b³= (a[Same sign]b)(a² [Opposite sign] ab [Always Positive] b²)

Signs of Factorization: Example

Signs of Factorization: Example

Geometrical Representation of Difference of Cubes

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The factorization of the difference of two cubes can be visualized geometrically as described follows:

Take two cubes, one with sides of length A and other with sides of length B as shown in the image below.

geometric representation

The expression A³- B³ can be viewed as the difference in volume between two cubes is equal to the remaining volume of the cube with side A after the other cube with side B has been cut out and removed from it.

expression

Note that each of the three highlighted edges has length A-B.

This shape can be cut into three rectangular prisms. Each prism has a side of length A-B.

Now, putting together the prisms in such a way that the entire shape has a depth of A-B.

Note that the volume of this object formed by putting the prisms together is (A-B) (A2+AB+B2)

We can now clearly visualize the expression of difference of cubes as seen in the image below:

Things to Remember

In case of difference of cubes,
- The first factor on the right side of the equation will be a binomial with a negative middle sign.
- The second factor will be a trinomial and it will always have a positive middle sign which is opposite to the middle sign of the expression on the left side of the equation.
To recognize the pattern of the difference of two cubes,
- Check whether the distribution has a binomial times a trinomial.
- The binomial being the difference between cube roots of the two terms representing cubes.
- The trinomial will contain the squares of the cube roots of the two terms and the opposite of their product.

Sample Questions

Ques. Prove that (a – b) (a² + ab + b²) =a³ – b³ . (3 Marks)

Ans. We start with the left hand side of the given expression, i.e.

(a – b) (a² + ab + b²)

Distributing (a-b) over the trinomial we get -

a (a² + ab + b²) - b (a² + ab + b²)

= (a³ + a²b + ab²) - (a²b+ ab² + b³)

= a³ + a²b + ab²- a²b - ab² - b³

= a³ + (a²b - a²b) + (ab² - ab²)- b³

= a³ - b³

Hence proved that (a – b) (a² + ab + b²) =a³ – b³

Ques. The expression x³-27 has two factors. If one factor is (x-3) then what will be the other factor?
a.(x³-3x+27)
b.(x²+3x+9)
c.(x²+3x-9)
d.(x²-9x+27)
e.(x²-3x+9) (3 Marks)

Ans. Option B

In the given binomial expression, there are two perfect cubes i.e. x³ and 3³

As per the formula of difference of cubes, the first factor will be a binomial and the second factor will be a trinomial.

a³ – b³ = (a – b) (a² + ab + b²)

The first factor has already been given here as the binomial (x-3)

To get the second factor we will replace a = x and b = 3 in the trinomial expression

(a² + ab + b²)

So the required trinomial is (x² + 3x + 3²) and the answer is Option B) (x² + 3x + 9)

Ques. Express (2-ab)(4+2ab+a²b²) as difference of cubes. (3 Marks)

Ans. The expression (2-ab)(4+2ab+a²b²) consists of a binomial and a trinomial and their product represents the difference of two perfect cubes.

The first factor, binomial (2-ab) consists of a number (2) and two variables (a,b)

The square of 2 = 4 and square of ab = a²b²

The cube of 2 = 8 and cube of ab = a³b³

So, (2-ab)(4+2ab+a²b²) = 2³ - a³b³

Ques. Factorize the expression 27x³- 125 by using the difference of cubes formula. (3 Marks)

Ans. As the two terms of the given expression 27x³ - 125 are perfect cubes

So, we get 27x³ - 125 = (3x)³ - 5³

Now, substitute a = 3x and b = 5 in the formula of the difference of cubes.

a³ - b³ = (a - b) (a² + ab + b²)

(3x)³ − 5³ = (3x − 5)((3x)² + (3x)(5) + 5²) = (3x − 5)(9x² + 15x + 25)

The answer is 27x³ - 125 = (3x - 5)(9x² + 15x + 25).

Ques. Find the value of 108³ - 8³ by using the difference of cubes formula. (3 Marks)

Ans. Let us assume that a = 108 and b = 8.

We will substitute these in the formula of difference of cubes.

a³ - b³ = (a - b) (a² + ab + b²)

108³−8³ = (108 − 8)(108² + (108)(8) + 8²) = (100)(11664 + 864 + 64) = (100)(12592) = 1259200

Answer is 108³ - 8³ = 1,259,200.

Ques. Factorise 8x⁶ - 125. (4 Marks)

Ans. 8x⁶ and 125 are perfect cubes, so we can apply the rule of difference of cubes to this polynomial expression.

To get the first factor for the difference of cubes, we will find the cube roots of the two terms of the given expression.

cube root of 8x⁶ is 2x²

cube root of 125 is 5

As the first factor is the difference of the cube roots, we get the binomial is (2x² - 5)

Now, we shall derive the trinomial using the binomial factor we just found.

To get the first term of the trinomial, square the first term 2x² to get 4x⁴

To get the middle term of the trinomial, multiply 2x²and 5 and change the sign from negative to positive to get +10x²

To get the third term of the trinomial, square the second term 5 to get 25

Putting these three terms together, the second factor trinomial is (4x⁴ + 10x² + 25)

So, the answer will be (2x - 5) (4x⁴ + 10x² + 25)

Ques. Factorize 250x⁴ - 128x. (4 Marks)

Ans. In the given expression, neither 250x⁴ nor 128x are perfect cubes

If we factor out the common factor 2x from this expression

this gives us 2x(125x³ - 64)

Now both 125x³ and 64 are perfect cubes

the cube root of 125x³ is 5x , and the cube root of 64 is 4

so our binomial is (5x - 4)

Now to get the trinomial we square 5x to get 25x² as our first term

we change the sign to + and multiply 5x and 4 to get +20x as the middle term

we square 4 to get 16 as our third term

so the trinomial is (25x² + 20x + 16) and the answer is 2x(5x - 4)(25x² + 20x + 16)

Ques. Factorize 40y³ – 625z³. (3 Marks)

Ans. To convert the given expression into a polynomial having perfect cubes we need to factor out the common factor from the two given terms

So, 40y³ – 625z³ = 5 (8y³ – 125z³)

Expressing the binomial as difference of cubes we get

8y³– 125z³ = (2y)³ - (5z)³

Applying the rule for difference of cubes, we can factorize this binomial as-

40y³ – 625z³ = 5 (8y³ – 125z³) = 5 [(2y)³ - (5z)³]

= 5 [ (2y-5z) ((2y)² + (2y)(5z) + (5z)²)]

= 5 [ (2y-5z) (4y²+10yz+25z²)]

Ques. Factorize - 1- 216a³b³. (2 Marks)

Ans. In the given expression, 1 = (1)(1)(1) = 1³ and 216 = 6³

1 - 216a³b³

= (1)³ - (6ab)³

= (1 - 6ab) [(1)² + (1)(6ab) + (6ab)²]

= (1 - 6ab) (1+6ab+36a²b²)

Also Read:

Introduction to Arithmetic Progression	Sum of Squares
Arithmetic Progression	Geometric Progression
Factorial Formula	Arithmetic Sequence
Sequence and Series Polynomials	Polynomials Formula Roots of Polynomials

Difference of Cubes Formula: Solved Examples

Difference of Cubes

Geometrical Representation of Difference of Cubes

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

2.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

5.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

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Difference of Cubes Formula: Solved Examples

Difference of Cubes

Geometrical Representation of Difference of Cubes

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

2. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

5.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

6.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

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SUBSCRIBE TO OUR NEWS LETTER

1.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

2.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

5.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

6.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).