Question

How do you find the remainder when \[f\left( x \right) = {x^4} + 8{x^3} + 12{x^2}\] is divided by \[x + 1\]?

Answer 1

Hint: Remainder theorem states that if a polynomial say \[h\left( y \right)\] is divided by a linear polynomial in the form \[y - a\], then the remainder for this division is equal to the value of polynomial \[h\left( y \right)\] at \[y = a\].

Complete step-by-step solution:
Apply the remainder theorem as divisor \[g\left( x \right) = x + 1\] is linear polynomial.
Evaluate \[g\left( x \right) = 0\] and obtain the value of \[x\]as shown below.
\[ \Rightarrow x + 1 = 0\]
\[ \Rightarrow x = - 1\]
Substitute \[ - 1\] for \[x\] in the polynomial \[f\left( x \right)\] and evaluate the value for that as shown below.
\[\begin{align}
&f\left( { - 1} \right) = {\left( { - 1} \right)^4} + 8{\left( { - 1} \right)^3} + 12{\left( { - 1} \right)^2}\\
& = 1 - 8 + 12\\
&= 5\\
\end{align}\]
Thus, from the remainder theorem it is observed that when \[f\left( x \right) = {x^4} + 8{x^3} + 12{x^2}\] is divided by \[g\left( x \right) = x + 1\] the remainder is \[5\].
The other way to obtain the remainder is by actual division method as shown below.
Arrange \[f\left( x \right)\] and \[g\left( x \right)\] in descending order of the power of the variable \[x\]and then divide as shown below.
\[\begin{array}{l}x + 1)\overline {{x^4} + 8{x^3} + 12{x^2}} ({x^3} + 7{x^2} + 5x - 5\\\,\,\,\,\,\,\,\,\,\,\,{x^4} + {x^3}\\\,\,\,\,\,\,\underline { - \,\,\,\,\,\,\,\, - \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7{x^3} + 12{x^2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,7{x^3} + 7{x^2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - \,\,\,\,\,\,\,\,\, - \,\,\,\,\,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5{x^2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5{x^2} + 5x\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - \,\,\,\,\,\,\,\,\,\, - \,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 5x\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 5x - 5\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline { - \,\,\,\,\,\,\, + \,\,\,\,} \\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,5\end{array}\]
Therefore, the quotient is \[{x^3} + 7{x^2} + 5x - 5\] and remainder is \[5\] when \[f\left( x \right) = {x^4} + 8{x^3} + 12{x^2}\] is divided by \[x + 1\].
It is observed from the actual division method that the remainder is still \[5\] same as it comes from the remainder theorem method.
Thus, the remainder is \[5\] when \[f\left( x \right) = {x^4} + 8{x^3} + 12{x^2}\] is divided by \[x + 1\]with the use of remainder theorem and it is verified by actual division method.

Note: For this type of problem students are encouraged to memorize the steps of remainder theorem as it is easy to use for obtaining the remainder for linear divisors. Also the actual division method is equally important as it is the basic way to find quotient and remainder. appropriate method should be selected on the basis of the degree of the given polynomial divisor.