Question

Divide $P\left( x \right)=3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2$ by $g\left( x \right)={{x}^{2}}+3x+1$ and find the quotient and remainder?

Answer 1

Hint: In this question, we have to divide the given polynomials to get the remainder. Thus, we will apply the long-division method. First, we will multiply the first term of the divisor by ${{x}^{2}}$ to make it equal to the first term of the dividend. Then, we will multiply the second term of the divisor by ${{x}^{2}}$ and place it below the second term of the dividend and change the sign of the new terms. Then, we will multiply the third term of the divisor by ${{x}^{2}}$and place it below the third term of the dividend and also change the sign of the new term. After, that we will make the necessary calculations and again apply the same steps as above, to get the required quotient and the remainder of the given problem.

Complete step by step answer:
According to the question, we have to divide the given polynomials.
Thus, we will use the long division method and the basic mathematical rules to get the solution.
We have to solve $P\left( x \right)=3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2$ by $g\left( x \right)={{x}^{2}}+3x+1$ ------------- (1)
Now, we know that both the numerator and the denominator of the given problem is in the form of polynomial, thus we will apply the long-division method in the algebraic fractional term, that is we get
$\Rightarrow {{x}^{2}}+3x+1\overset{{}}{\overline{\left){3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2}\right.}}$
Now, we know that the divisor have the first term as ${{x}^{2}}$ and the first term of the dividend is $3{{x}^{4}}$ , thus if we multiply the first term of the divisor with the $3{{x}^{2}}$ , we will get the first term of the dividend. The second and third term of the divisor is multiplied by the quotient which is to be placed below the second term of the dividend, and we will change the sign, thus we get
\[\Rightarrow {{x}^{2}}+3x+1\overset{3{{x}^{2}}}{\overline{\left){\begin{align}
  & 3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2 \\
 & \underline{{}_{-}3{{x}^{4}}\underset{-}{\mathop{+9}}\,{{x}^{3}}\underset{-}{\mathop{+}}\,3{{x}^{2}}} \\
\end{align}}\right.}}\]
On solving the above division, we get
$\Rightarrow {{x}^{2}}+3x+1\overset{3{{x}^{2}}}{\overline{\left){\begin{align}
  & 3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2 \\
 & \underline{{}_{-}3{{x}^{4}}\underset{-}{\mathop{+9}}\,{{x}^{3}}\underset{-}{\mathop{+}}\,3{{x}^{2}}} \\
 & \text{ }-4{{x}^{3}}-10{{x}^{2}} \\
\end{align}}\right.}}$
Now, we will again apply the same method as done in the above step, we get
$\Rightarrow {{x}^{2}}+3x+1\overset{3{{x}^{2}}-4x}{\overline{\left){\begin{align}
  & 3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2 \\
 & \underline{{}_{-}3{{x}^{4}}\underset{-}{\mathop{+9}}\,{{x}^{3}}\underset{-}{\mathop{+}}\,3{{x}^{2}}} \\
 & \text{ }-4{{x}^{3}}-10{{x}^{2}}+2x+2 \\
 & \text{ }\underline{\underset{+}{\mathop{-4}}\,{{x}^{3}}\underset{+}{\mathop{-}}\,12{{x}^{2}}\underset{+}{\mathop{-4}}\,x} \\
 & \text{ +2}{{x}^{2}}+6x \\
\end{align}}\right.}}$
In the last, we will again apply the same method as done in the above step, we get
$\Rightarrow {{x}^{2}}+3x+1\overset{3{{x}^{2}}-4x+2}{\overline{\left){\begin{align}
  & 3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2 \\
 & \underline{{}_{-}3{{x}^{4}}\underset{-}{\mathop{+9}}\,{{x}^{3}}\underset{-}{\mathop{+}}\,3{{x}^{2}}} \\
 & \text{ }-4{{x}^{3}}-10{{x}^{2}}+2x+2 \\
 & \text{ }\underline{\underset{+}{\mathop{-4}}\,{{x}^{3}}\underset{+}{\mathop{-}}\,12{{x}^{2}}\underset{+}{\mathop{-4}}\,x} \\
 & \text{ +2}{{x}^{2}}+6x+2 \\
 & \text{ }\underline{\underset{-}{\mathop{+}}\,2{{x}^{2}}\underset{-}{\mathop{+}}\,6x\underset{-}{\mathop{+}}\,2} \\
 & \text{ }\underline{\underline{\text{ 0}}} \\
\end{align}}\right.}}$
Thus, we get remainder is equal to 0 and the quotient is equal to $3{{x}^{2}}-4x+2$ .

Therefore, if we divide $P\left( x \right)=3{{x}^{4}}+5{{x}^{3}}-7{{x}^{2}}+2x+2$ by $g\left( x \right)={{x}^{2}}+3x+1$ , then its quotient is equal to $3{{x}^{2}}-4x+2$ and remainder is equal to 0.

Note: While solving this problem, do all the calculations properly to avoid confusion and error. You can also check your answer, using the formula $dividend=\left( divisor\times quotient \right)+remainder$ , that is simply we put the value of divisor, quotient, and the remainder in the above equation, to get the dividend.