Question

The quotient and remainder when \[3{{x}^{4}}+6{{x}^{3}}-6{{x}^{2}}+2x-7\] is divided by \[x-3\] are
(A) Quotient: \[3{{x}^{3}}+15{{x}^{2}}+39x+119\] and Remainder: 350
(B) Quotient: \[3{{x}^{3}}+10{{x}^{2}}+39x+119\] and Remainder: 35
(C) Quotient: \[3{{x}^{3}}+15{{x}^{2}}+39x+119\] and Remainder: 50
(D) Quotient: \[3{{x}^{3}}+15{{x}^{2}}+119\] and Remainder: 350

Answer 1

Hint: In the above question, we are asked to find the quotient and the remainder by dividing the algebraic expression \[3{{x}^{4}}+6{{x}^{3}}-6{{x}^{2}}+2x-7\] by the monomial \[x-3\]. We will carry out the long division method to find the quotient and the remainder of the above given operation. Hence, we will have the value of the quotient and the remainder of the given division.

Complete step-by-step answer:
According to the given question, we are asked to find the quotient and the remainder by dividing \[3{{x}^{4}}+6{{x}^{3}}-6{{x}^{2}}+2x-7\] by \[x-3\]. We will have to carry out the division operation to find the same.
We will be using the long division method to compute the division operation.
The dividend given to us is a polynomial of degree 4 and that is, \[3{{x}^{4}}+6{{x}^{3}}-6{{x}^{2}}+2x-7\] and the divisor given to us is a monomial and we know that the monomial has a degree of 1, that is, \[x-3\].
We will divide the polynomial expression with the monomial expression and we get,
\[x-3\overset{3{{x}^{3}}+15{{x}^{2}}+39x+119}{\overline{\left){\begin{align}
  & 3{{x}^{4}}+6{{x}^{3}}-6{{x}^{2}}+2x-7 \\
 & \underline{-(3{{x}^{4}}-9{{x}^{3}})} \\
 & 0{{x}^{4}}+15{{x}^{3}}-6{{x}^{2}} \\
 & \underline{-(15{{x}^{3}}-45{{x}^{2}})} \\
 & 0{{x}^{3}}+39{{x}^{2}}+2x-7 \\
 & \underline{-(39{{x}^{2}}-117x)} \\
 & 0{{x}^{2}}+119x-7 \\
 & \underline{-(119x-357)} \\
 & \_\_\_\_\_\_350\_\_ \\
\end{align}}\right.}}\]
As seen from the above division operation, we can conclude that the quotient that we obtained from the above division operation, we get, \[3{{x}^{3}}+15{{x}^{2}}+39x+119\]. And the remainder that we obtained is \[350\].
Therefore, option (A) Quotient: \[3{{x}^{3}}+15{{x}^{2}}+39x+119\] and Remainder: 350 is the correct answer.

So, the correct answer is “Option A”.

Note: The division operation should be done clearly and step-wise without missing any terms or overlooking any term. The quotient is the expression which when multiplied with the divisor gives the dividend when the remainder is zero. In the division process, the degree of the quotient will always be less than that of the degree of dividend.