Question

Find the divisor of ${{x}^{2}}+1$ which gives the quotient as x+1 and leaves 2 as the remainder.

Answer 1

Hint: Here we are given the value of dividend, remainder and quotient. We need to find out the value of the divisor. A divisor is a number or an algebraic equation that divides another number or algebraic equation either completely or with a remainder.
Use the relation,
Dividend = Divisor$\times $Quotient + Remainder
Formula Used:
Dividend = Divisor$\times $Quotient + Remainder
${{x}^{2}}-1=(x-1)(x+1)$

Complete step-by-step answer:
Dividend = ${{x}^{2}}+1$, Quotient = x+1 and Remainder = 2 is given in the question and we need to find out the divisor.
We know that,
Dividend = Divisor$\times $Quotient + Remainder (1)
Let divisor = D (2)
Put the values given the question in equation (1)
We get,
$\Rightarrow $ ${{x}^{2}}+1$ =D (x+1) + 2
$\Rightarrow $${{x}^{2}}+1-2=D(x+1)$
$\Rightarrow {{x}^{2}}-1=D(x+1)$
Using the identity ${{x}^{2}}-1=(x-1)(x+1)$
$\Rightarrow (x-1)(x+1)=D(x+1)$
Here (x+1) will be cancelled from both the sides
We get,
$\Rightarrow x-1=D$
D= divisor (from equation (2))
$\Rightarrow $ the divisor, D= x-1 is the answer.

Additional information:
In division we will see the relationship between the dividend, divisor, quotient and remainder. The number which we divide is called the dividend. The number by which we divide is called the divisor. The result obtained is called the quotient. The number left over is called the remainder.
Some Binomial Theorem identities are-
\[\begin{align}
  & {{x}^{2}}-{{y}^{2}}=(x+y)(x-y) \\
 & {{(x+y)}^{2}}={{x}^{2}}+2xy+{{y}^{2}} \\
 & {{(x-y)}^{2}}={{x}^{2}}-2xy+{{y}^{2}} \\
 & {{(x+y)}^{3}}={{x}^{3}}+3{{x}^{2}}y+3x{{y}^{2}}+{{y}^{3}} \\
\end{align}\]
\[\begin{align}
  & {{(x-y)}^{3}}={{x}^{3}}-3{{x}^{2}}y+3x{{y}^{2}}-{{y}^{3}} \\
 & {{x}^{3}}+{{y}^{3}}=(x+y)({{x}^{2}}-xy+{{y}^{2}}) \\
 & {{x}^{3}}-{{y}^{3}}=(x-y)({{x}^{2}}+xy+{{y}^{2}}) \\
\end{align}\]

Note: The knowledge of the relation between quotient, remainder, dividend and divisor is required to answer this question. Additional information of some binomial theorem identities is also required to answer this question.