Question

factorise
a^3 + 2a^2 - a - 2​

Answer 1

Given:

The expression -

• $\bf \to a^3 + 2a^2 - a - 2$

What To Find:

We have to -

• Factorise the given expression.

Solution:

$\bf \to a^3 + 2a^2 - a - 2$

Regroup the given terms,

$\bf \to (a^3 - a) + (2a^2- 2)$

Where -

• $\bf \to (a^3 - a) - [1st \: Term]$

• $\bf \to (2a^2- 2) - [2nd \: term]$

Find out the common factor of the first term,

$\bf \to (a^3 - a) : a$

Take the common factor out of the bracket,

$\bf \to a(a^2 - 1)$

Find out the common factor of the second term,

$\bf \to (2a^2- 2) : 2$

Take the common factor out of the bracket,

$\bf \to 2(a^2 - 1)$

Put it in the expression,

$\bf \to a(a^2 -1 ) + 2(a^2- 1)$

Take the brackets as a common factor and take a and 2 as one factor,

$\bf \to (a^2- 1) (a+ 2)$

Verification:

We have to bring it in the below way,

$\bf \to (a^2- 1) (a+ 2) = a^3 + 2a^2 - a - 2$

Thus, we will solve the LHS,

$\bf \to (a^2- 1) (a+ 2)$

Can be written as,

$\bf \to a^2(a+ 2) - 1(a+ 2)$

Solve the first brackets,

$\bf \to a^3+ 2a^2 - 1(a+ 2)$

Solve the second brackets,

$\bf \to a^3+ 2a^2 - a-2$

Compare with RHS,

$\bf \to a^3+ 2a^2 - a-2 = a^3+ 2a^2 - a-2$

∵ LHS = RHS

∴ Hence, verified.

Final Answer:

∴ Thus, the factors of the expression $\bf a^3 + 2a^2 - a - 2$ are $\bf (a^2- 1)$ and $\bf (a+ 2)$.