Question

Factories- i) a³ +27b³

Answer 1

${a}^{3} + (3b) ^{3} \\ = (a + b)( {a}^{2} - 3ab + 9 {b}^{2} )$

Answer 2

Answer:

$\bold{(a+3b)(a^2 - 3ab + 9b^2)}$

Step-by-step explanation:

This question will be solved using the algebraic identity a³ + b³.

So, let's derive the identity for a³ + b³.

For this, we need to start from (a + b)³.

We know that

$(a+b)^3 = a^3 + b^3 +3ab(a+b)$

So,

$a^3 + b^3 = (a+b)^3 - 3ab(a+b)$

This can be simplified to the expression below.

$(a + b)(a^2 - ab + b^2)$

Thus, we have,

$\bold{a^3+b^3 = (a + b)(a^2 - ab + b^2)}$

Now, coming to your question,

a³ +27b³ can be simplified as below.

$(a)^3 +(3b)^3$

Because 3³ = 27.

Using the identity,

$(a)^3 +(3b)^3 = (a+3b)((a^2) - (a)(3b) + (3b)^2)\\\\= (a+3b)(a^2 - 3ab + 9b^2)$

Hence, the answer to your question is as below:

$\bold{(a+3b)(a^2 - 3ab + 9b^2)}$