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Sum of Cubes Formula

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  • Last Updated : 16 Mar, 2022

The sum of cubes formula is used while adding up any two polynomials, a3 + b3. When solving algebraic equations of various sorts, this factorization formula comes in helpful. This formula is also simple to remember and may be done in a couple of minutes. It works in a similar way to the difference in cubed formula. It is discussed as follows.

Sum of Cubes

a3 + b3 = (a + b)(a2 – ab + b2)

where, a and b are any two given variables.

Proof

LHS = a3 + b3

RHS = (a + b)(a2 – ab + b2)

Using the distributive property of multiplication, we have:

RHS = 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

Hence proved.

Sample Questions

Question 1: Factorize 343a3 + 216 using sum of cubes.

Solution:

343a3 + 216 = (7a)3 + (6)3

Since, a3 + b3 = (a + b)(a2 – ab + b2)

(7a)3 + (6)3 = (7a + 6)[(7a)2 – (7a)(6) + (6)2]

= (7a + 6)[49a2 – 42a + 36]

Question 2: Factorize 8p3 + 27 using sum of cubes.

Solution:

8p3 + 27 = (2p)3 + (3)3

Since, a3 + b3 = (a + b)(a2 – ab + b2)

(2p)3 + (3)3 = (2p + 3)[(2p)2 – (2p)(3) + (3)2]

= (2p + 3)[4p2 – 6p + 9]

Question 3: Factorize 27t3 + 125 using the sum of cubes.

Solution:

27t3 + 125 = (3t)3 + (5)3

Since, a3 + b3 = (a + b)(a2 – ab + b2)

(3t)3 + (5)3 = (3t + 5)[(3t)2 – (3t)(5) + (5)2]

= (3t + 5)[9t2 – 15t + 25]

Question 4: Factorize 64s3 + 125 using sum of cubes.

Solution:

64s3 + 125 = (8s)3 + (5)3

Since, a3 + b3 = (a + b)(a2 – ab + b2)

(8s)3 + (5)3 = (8s + 5)[(8s)2 – (8s)(5) + (5)2]

= (8s + 5)[64s2 – 40s + 25]

Question 5: Factorize 512 + 729v3 using the sum of cubes formula.

Solution:

512 + 729v3 = (8)3 + (9v)3

Since, a3 + b3 = (a + b)(a2 – ab + b2)

(8)3 + (9v)3 = (8 + 9v)[(8)2 – (8)(9v) + (9v)2]

= (8 + 9v)[64 – 72v + 729v2]

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