Question

The polynomial which when divided by$-{{x}^{2}}+x-1$ gives a quotient x - 2 and remainder is 3, is
A.${{x}^{3}}-3{{x}^{2}}+3x-5$
B.$-{{x}^{3}}-3{{x}^{2}}-3x-5$
C.$-{{x}^{3}}+3{{x}^{2}}-3x+5$
D.${{x}^{3}}-3{{x}^{2}}-3x+5$

Answer 1

Hint: In this problem, we are required to calculate the polynomial. We are given a divisor, quotient and remainder. So, by using the basic definition of dividend we can easily evaluate the required polynomial.

Complete step-by-step answer:
According to the problem statement, we are given a divisor $-{{x}^{2}}+x-1$ of a polynomial. This divisor when divided from the dividend polynomial leaves x - 2 as quotient and 3 as remainder. We are required to find the final expression of a polynomial.
As we know that, the product of divisor and quotient add to the remainder gives the dividend of a number or an expression. This can be mathematically expressed as:
Let f(x) be the dividend polynomial having a divisor p(x) and a quotient q(x). The polynomial also leaves a remainder r(x) upon division. Therefore,
$f(x)=p(x)\cdot q(x)+r(x)$
By using this basic definition of dividend, and putting $p(x)=-{{x}^{2}}+x-1,\text{ }q(x)=x-2\text{ and }r(x)=3$, we get
$\begin{align}
  & f(x)=\left( -{{x}^{2}}+x-1 \right)\cdot \left( x-2 \right)+3 \\
 & f(x)=\left( -{{x}^{2}} \right)\cdot \left( x-2 \right)+x\cdot \left( x-2 \right)-1\cdot \left( x-2 \right)+3 \\
 & f(x)=-{{x}^{3}}+2{{x}^{2}}+{{x}^{2}}-2x-x+2+3 \\
 & f(x)=-{{x}^{3}}+3{{x}^{2}}-3x+5 \\
\end{align}$
So, the required polynomial is $-{{x}^{3}}+3{{x}^{2}}-3x+5$.
Therefore, option (c) is the correct answer.

Note: This is a direct problem which can be easily solved by the knowledge of expression of dividend in terms of divisor, quotient and remainder. Students must take care while multiplying the exponents of x and the negative sign respective to each term as silly mistakes due to calculation error are bound to occur.