Question

Find the remainder when the term $\left( {{{\text{x}}^3} + 1} \right)$ is divided by the term $\left( {{\text{x + 1}}} \right)$.

Answer 1

Hint: To find the remainder,first expand the following given term by using the formula of $\left( {{{\text{a}}^3} + {{\text{b}}^3}} \right)$= $\left( {{\text{a + b}}} \right)\left( {{{\text{a}}^2} + {\text{ab + }}{{\text{b}}^2}} \right)$ and then use the formula Dividend = divisor x quotient + remainder to approach at the answer.

Complete step-by-step answer:
$\left( {{{\text{x}}^3} + 1} \right)$ can be written as $\left( {{{\text{x}}^3} + 1} \right)$ = $\left( {{{\text{x}}^3} + {1^3}} \right)$

This is in the form of $\left( {{{\text{a}}^3} + {{\text{b}}^3}} \right)$.

We know, $\left( {{{\text{a}}^3} + {{\text{b}}^3}} \right)$= $\left( {{\text{a + b}}} \right)\left( {{{\text{a}}^2} + {\text{ab + }}{{\text{b}}^2}} \right)$
Comparing with the given term we have,
$ \Rightarrow \left( {{{\text{x}}^3} + 1} \right) = \left( {{\text{x + 1}}} \right)\left( {{{\text{x}}^2} + {\text{x}} + 1} \right)$

We know that,
Dividend = divisor x quotient + remainder.

Comparing $\left( {{{\text{x}}^3} + 1} \right) = \left( {{\text{x + 1}}} \right)\left( {{{\text{x}}^2} + {\text{x}} + 1} \right)$with the above formula.
Here, Dividend = $\left( {{{\text{x}}^3} + 1} \right)$
            Divisor = $\left( {{\text{x + 1}}} \right)$
          Quotient = $\left( {{{\text{x}}^2} + {\text{x + 1}}} \right)$
There is no term in place of remainder which means it is 0.

Hence, remainder is 0.

Note: In order to solve such types of problems the key is to have adequate knowledge in algebraic formulas and identities is required in order to pick a perfect formula to simplify the given question. Dividend – the terms which are to be divided, Divisor – the term with which we divide, Quotient – the result obtained by dividing the dividend with the divisor and remainder – the leftover part after the others have been dealt with.