Sum of Cubes Formula: Definition, Proof & Solved Examples

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Sum of Cubes Formula is the formula used to find the result of two polynomials upon addition (a³+ b³). The sum of cubes formula is derived by calculating the area of a particular region by using two different methods. The first is to square a side’s length and then add the smaller squares’ areas. It can be said alternatively that the first n natural numbers’ sum will be the sum of the first n cubes. The sum of cubes formula proves to be very useful while solving algebraic expressions of various types. The formula is quite similar to the difference in cubes formula as well.

Table of Content

Sum of Cubes Formula
Proof For Sum of Cubes Formula
How to Solve a Question With Sum of Cubes Formula
Things to Remember
Sample Questions

Key Terms: Sum of Cubes Formula, Cube, Variable, Multiplication, Power, Algebraic Identities, Polynomial, Factoring Formula

Also Read: Mathematics Preparation Tips

Sum of Cubes Formula

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The formula of the sum of cubes is also known as the factoring formula. The following formula is used to determine the sum of cubes of any polynomial:

a³ + b³ = (a + b)(a² - ab + b²)

Sum of Cubes Formula

Sum of Cubes Formula

Here, a is the first variable, and b is the second variable.

Also Read: Sum of Squares Formula

Proof For Sum of Cubes Formula

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The proof or the verification of the formula of the sum of cubes {a³ + b³ = (a + b) (a² - ab + b²} is given by proving that LHS = RHS here.

LHS = a³ + b³

After the solution of RHS term, we find,

= (a + b) (a² - ab + b²)

After the multiplication of a and b with (a² + ab + b²), we get

= a (a² - ab + b²) + b(a² - ab + b²)

= a³ - a²b + ab² + a²b - ab² + b³

= a³ - a²b + a²b + ab²- ab² + b³

= a³ - 0 + 0 + b³

= a³ + b³

LHS = RHS

Hence, the Sum of Cubes Formula is proved.

Also Read: Statistics

How to Solve a Question With Sum of Cubes Formula?

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For solving a question with the sum of cubes formula, we shall use the following steps.

Take note of the pattern of the two numbers and check if they have the power of 3 on them or not.
Now, note down the formula for the sum of cubes that is a³ + b³ = (a + b) (a² - ab + b²)
Substitute the values of b and a in the formula of the sum of cubes and simplify it.

For Example: Find the value of (100³ + 2³) using the formula for the sum of cubes.

Solution: To Find: 100³ + 2³

Let's take a = 100 and b = 2.

Substitute these values in the formula of the sum of cubes that is, a³ + b³

a³+ b³ = (a + b) (a² - ab + b²)

=100³+2³ = (100+2)(1002 - (100)(2)+22)

= (102) (10000-200+4)

= (102)(9804)

= 1000008

Therefore the Answer is: 100³ + 2³ = 1000008.

Things to Remember

The formula for the Sum of the cubes is the formula that is used to find the resultant of the addition of two polynomials (a³+ b³).
The formula of the sum of cubes is also known as the factoring formula.
The extended form of the formula is a³ + b³ = (a + b)(a² - ab + b²). Here, b is the second variable, and a is the first variable.
The values given in a question should have a power of 3 upon them on each variable.

Also Read: Distance Time Graph

Solved Questions

Ques. Simplify the following equation 19³+ 20³ with the use of the sum of cubes formula. (3 marks)

Ans. To find 19³ + 20³

Let a = 19 and b = 20

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

a³ + b³ = (a + b) (a² - ab + b²)

=19³+20³ = (19+20)(192 - (19)(20)+202)

= (39)(361-380+400)

= (39)(381)

= 14,859

Therefore the answer is: 19³ + 20³ = 14859.

Ques. Factorise the following equation 27x³ + 1. (3 marks)

Ans. To factorise 27x³ + 1

Let a = 3x and b = 1

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

= 27x³ + 1 = (3 x)³ + 13

= (3 x+1)[(3 x)² − (3 x)(1) + 12)]

= (3 x + 1)(9 x² – 3 x + 1)

Ques. Factorise the following equation: 8x³ +125 (3 marks)

Ans. To factorise: 8x³ + 125

Let a = 2x and b = 5

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

=8x³ + 125 = (2x)³ + 53

a³ + b³ = (a + b)(a² – ab + b²)

Therefore, 8x³ + 125 = (2x + 5)[(2x)² – (2x)(5) + 52] = (2x + 5)(4x² – 10x + 25)

Ques. Factorise the following equation: 64x³ + 27y³ (3 marks)

Ans. To factorise: 64x³ + 27y³

64x³ + 27y³ = (4x)³ + (3y)³

Let a = 4x and b = 3y

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

a³ + b³ = (a + b)(a² – ab + b²)

Therefore, 64x³ + 27y³ = (4x + 3y)[(4x)² – (4x)(3y) + (3y)²] = (4x + 3y)(16x² – 12xy + 9y²)

Ques. Factorise the following equation: x³ – 8. (3 marks)

Ans. To factorise: x³ - 8.

x³ – 8 = x³ – 23

Let a = x and b = 2

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

= (x – 2)(x² + 2x + 22)

= (x – 2)(x² + 2x + 4)

Ques. Factorise the following equation: 27x³ + 1 (3 marks)

Ans. To factorise: 27x³ + 1

27x³ + 1 = (3x)³ + 13

Let a = 3x and b = 1

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

= (3x + 1)((3x)² – (3x)(1) + 12)

= (3x + 1)(9x² – 3x + 1)

Ques. Factorise the following equation x³y⁶+ 64. (3 marks)

Ans. X³y⁶ + 64 = (xy²)³ – 43

Let a = xy.2 and b = 4

By Using the sum of cubes formula a³ + b³ = (a + b) (a² - ab + b²)

We now substitute the values in the a³ + b³ formula

x³y⁶ – 64 = (xy²)³ – 43

= (xy² – 4)((xy²)² + (xy²)(4) + 42)

= (xy² – 4)(x²y⁴ + 4xy² + 16)

Ques. If the value of a³ + b³ = 405 and the value of a + b = 9, then find the value of ab. (3 marks)

Ans. Given: a³ + b³ = 405

→ a + b = 9

The Formula to be used: (a + b)³ = a³ + b³ + 3ab(a + b)

Now, Substituting the values in the formula

⇒ (9)³ = 405 + 3ab (9)

⇒ 729 = 405 + 27ab

⇒ 27ab = 324

⇒ ab = 12

Therefore, The value of ab is found to be 12.

Ques. Find the value of a³ + b³, if a/b + b/a = 1. (3 marks)

Ans. To find: a³ + b³

Since, a/b + b/a = 1,

So, (a² + b²)/ab = 1 → a² + b² = ab

Now, a² + b² = ab (eq 1)

= a² = ab - b²

= a² = b(a-b)

= a².a= a.b(a-b)

= a³ = ab(a-b) (eq 2)

Using eq 1 we can write,

b²= ab - a²

b².b = b.a(b-a)

b³ = ab(b-a) (eq 3)

So, a³ + b³ = ab(a-b) + ab(b-a) (by using eq 2 and eq 3)

a³+ b³ = ab

Therefore, a³ + b³ = 0.

Ques. Use the sum of cubes formula to find the factor of 216x³ + 64. (3 marks)

Ans. To find: The Factor of 216x³ + 64 by using the sum of cubes formula.

216x³+ 64 = (6x)³ + 43

Now, Use the sum of cubes formula,

a³ + b³ = (a + b)(a² - ab + b²)

Putting the particular values,

(6x)³ + 43 = (6x + 4)((6x)² - 6x × 4 + 42)

(6x)³ + 43 = (6x + 4)(36x² - 24x +16)

(6x)³ + 43 = 8(3x + 2)(9x² - 6x + 4)

Therefore, the factor of 216x³ + 64 is 2(3x + 2)(9x² - 6x + 4).

Ques. Find the factor of 8x³ + 125y³. (3 marks)

Ans. To find: The Factor of 8x³ + 125y³ by using the sum of cubes formula.

8x³ + 125y³ = (2x)³ + (5y)³

Now Use the sum of cubes formula,

a³+b³ = (a + b)(a² - ab + b²)

Putting the required values,

(2x)³ + (5y)³ = (2x + 5y)((2x)² – (2x)(5y) + (5y)²)

(2x)³ + (5y)³ = (2x + 5y)(4x² – 10xy + 25y²)

Therefore, the factor of 8x³ + 125y³ is (2x + 5y)(4x² – 10xy + 25y²).

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Sum of Cubes Formula: Definition, Proof & Solved Examples

Sum of Cubes Formula

Proof For Sum of Cubes Formula

How to Solve a Question With Sum of Cubes Formula?

Things to Remember

Solved Questions

CBSE CLASS XII Related Questions

1.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

2.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

3.
Find:

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

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
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.

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

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Sum of Cubes Formula: Definition, Proof & Solved Examples

Sum of Cubes Formula

Proof For Sum of Cubes Formula

How to Solve a Question With Sum of Cubes Formula?

Things to Remember

Solved Questions

CBSE CLASS XII Related Questions

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

2. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

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

4. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.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.

6.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

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SUBSCRIBE TO OUR NEWS LETTER

1.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

2.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

3.
Find:

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

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

5.
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.

6.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).