Question

Find the quotient and remainder when $p\left( x \right) = {x^4} + {x^3} - 3{x^2} + 17x - 8$ is divided by $g\left( x \right) = x - 5$.

Answer 1

Hint:In the given question, we are required to divide the polynomial $p\left( x \right) = {x^4} + {x^3} - 3{x^2} + 17x - 8$ by another polynomial $g\left( x \right) = x - 5$ to find the quotient and remainder in the division process. So, we will use the long division algorithm for division of $p\left( x \right) = {x^4} + {x^3} - 3{x^2} + 17x - 8$ by $g\left( x \right) = x - 5$ where $p\left( x \right)$ is dividend and $g\left( x \right)$ is divisor and we will obtain $q\left( x \right)$ as quotient and $r\left( x \right)$ as remainder.

Complete answer:
So, in the problem, dividend $ = p\left( x \right) = {x^4} + {x^3} - 3{x^2} + 17x - 8$
Divisor $ = g\left( x \right) = x - 5$
Now, we will follow the long division method to divide one polynomial by another and to find the quotient and remainder.
\[x - 5\overset{{x^3}+6x^2+27x+152}{\overline{\left){\begin{align}
  & {{x}^{4}}+x^3-3{{x}^{2}}+17x-8 \\
 & \underline{{{x}^{4}}-5{x^3}} \\
 & 6{{x}^{3}}-3x^2+17x-8 \\
 & \underline{6{{x}^3}-30x^2} \\
 & 27x^2+17x-8 \\
 & \underline{27{{x}^2}-135x} \\
 & 152x-8 \\
 & \underline{152x-760} \\
 & 752\\
\end{align}}\right.}}\]
Hence, we have divided the polynomial $p\left( x \right) = {x^4} + {x^3} - 3{x^2} + 17x - 8$ by $g\left( x \right) = x - 5$ using the long division method and obtained the quotient polynomial as $q\left( x \right) = {x^3} + 6{x^2} + 27x + 152$ and remainder polynomial as $r\left( x \right) = 752$.

Now, we can also verify this division method by using the property $\text{Dividend = Divisor x Quotient + Remainder}$.
$ \Rightarrow {x^4} + {x^3} - 3{x^2} + 17x - 8 = \left( {x - 5} \right) \times \left( {{x^3} + 6{x^2} + 27x + 152} \right) + 752$
Finding the product of polynomials, we get,
$ \Rightarrow {x^4} + {x^3} - 3{x^2} + 17x - 8 = \left( {{x^4} + 6{x^3} + 27{x^2} + 152x} \right) - \left( {5{x^3} + 30{x^2} + 135x + 760} \right) + 752$
Adding up the like terms on right side of equation, we get,
\[ \Rightarrow {x^4} + {x^3} - 3{x^2} + 17x - 8 = {x^4} + 6{x^3} + 27{x^2} + 152x - 5{x^3} - 30{x^2} - 135x - 760 + 752\]
\[ \therefore {x^4} + {x^3} - 3{x^2} + 17x - 8 = {x^4} + {x^3} - 3{x^2} + 17x - 8\]
Since both the sides of the equation are equal. So, the division process is done correctly.

Hence, the quotient is equal to $q\left( x \right) = {x^3} + 6{x^2} + 27x + 152$ and remainder polynomial is equal to $r\left( x \right) = 752$.

Note:The property of long division $Dividend = Divisor \times Quotient + Remainder$ holds true for division of numbers as well as polynomials. The degree of dividend given to us is $4$ and the degree of the divisor is $1$. So, we must remember that the degree of the remainder must be less than that of the divisor. So, in this case, the degree of remainder obtained is $0$ as $752$ is a constant polynomial.