What Is the Expansion of a3 + b3 Formula?

a3 + b3 = (a + b) (a2 - ab + b2)

What Is the a3 + b3 Formula in Algebra?

a3 + b3 = (a + b) (a2 - ab + b2)

A cube plus B cube formula

What is the formula of A cube plus B cube

Table of Contents

A cube plus B cube formula is an algebraic identity which is used to find the sum of the two cubes. The expression of this A cube plus B cube formula is a³+ b³= (a + b) (a² – ab + b²)

How to derive A cube plus B cube ka formula

Lets understand this a³+ b³ Formula in details. this formula is the sum of the two cubes. Here we will see the other expressions of A cube plus B cube ka formula or you can say other forms of a³+ b³ formula.

We know already the formula of (a+b)³ Here we will use this A plus B ka whole cube formula to prove A cube plus B cube ka formula. so (a+b)³=a³+b³+3ab(a+b).

(a+b)³=a³+b³+3ab(a+b)

a³+b³ = (a+b)³–3ab(a+b)

a³+b³=(a+b)³–3ab(a+b) [Note: Take comman part (a+b)]

a³+b³=(a+b)((a+b)²–3ab)

We know that formula of A plus B ka whole square (a+b)² = a²+b²+2ab

a³+b³=(a+b)(a²+b²+2ab–3ab)

a³+b³=(a+b)(a²+b²–ab)

Proof of a³+ b³ Formula

Now we will prove whether this A cube plus B cube a³+ b³= (a + b) (a² – ab + b²) formula is right or wrong ? for that we need to prove Left hand side (LHS) = Right hand side (RHS). Lets begin with the following steps.

LHS = a³+ b³
On Solving RHS side we get,
= (a + b) (a² – ab + b²)
On multiplying the a and b separately 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
Hence proved, LHS = RHS

List of Some important Algebraic formula

(a – b)³ = a³ – b³ – 3ab (a – b)
(a + b)³ = a³ + b³ + 3ab (a + b)
(a + b)² = a² + 2ab + b²
(a – b)² = a² – 2ab + b²
(a + b) (a – b) = a² – b²
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
(a – b)⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴
(x + y + z)² = x² + y² + z² + 2xy +2yz + 2xz
(x + y – z)² = x² + y² + z² + 2xy – 2yz – 2xz
(x – y + z)² = x² + y² + z² – 2xy – 2yz + 2xz
(x – y – z)² = x² + y² + z² – 2xy + 2yz – 2xz
x³ + y³ + z³ – 3xyz = (x + y + z) (x² + y² + z² – xy – yz -xz)
x³ + y³ = (x + y) (x²– xy + y²)
x³ – y³ = (x – y) (x²+ xy + y²)

Solved examples using a³ + b³ = (a + b) (a² – ab + b²) formula

Now you can understand this a³ + b³ = (a + b) (a² – ab + b²) formula in detail by solving some good examples

Example 1: Find the value of 102³ + 8³by using the A cube plus B cube a³+b³ formula.

Solution: In this question we have to to find: 102³ + 8³. Here we will assume that a = 102 and b = 8. So we will replace these A and B with 102 and 8 respectively in the formula of a³+ b³= (a + b) (a² – ab + b²).

102³+8³ = (102+8)(102² – (102)(8)+8²)
= (110) (10404-816+64)
= (110)(9652)
=1061720
Answer: 102³ + 8³ = 1,061,720.

Example 2: Factorize the following expression 8p³ + 27 by using the A cube plus B cube a³+b³ = (a + b) (a² – ab + b²) formula.

Solution: Here we have to factorize: 8p³ + 27. We will use the a³ + b³formula to factorize this expression. Lets assume a = 2p and b = 3. so we can rewrite the given expression in the form of a³ + b³

8p³ + 27 = (2p)³ + 3³
a³+ b³= (a + b) (a² – ab + b²)
(2p)³+3³ =(2p+3)((2p)²-(2p)(3)+3²)
= (2p+3) (4p²-6p+9)

Example 3: Factorize the following expression : 8p³ + 27q³

Solution : Write (8p³ + 27q³) in the form of (a³ + b³). 8p³ + 27p³ = (2p)³ + (3q)³

(2p)³ + (3q)³ is in the form of (a³+ b³). After comparing (a³+ b³) and (2p)³+ (3q)³, we get a = 2p and b = 3q.

Now we will solve this question this the help of (a³+ b³) formula. a³ + b³= (a + b)(a² – ab + b²)

Replace a and b with 2p and 3q respectively.

(2p)³ + (3q)³ = (2p + 3q)[(2p)² – (2p)(3q) + (3q)²]
8p³ + 27q³ = (2p + 3q)(8p² – 6pq + 9q²)