Chapter 13.6, Problem 50E

To determine

To find: the remainder using remainder theorem and state whether the binomial is factor of the polynomial or not.

Answer to Problem 50E

The remainder is $0$ and the binomial is factor of the polynomial.

Explanation of Solution

Given:

The divisor is $x+4$ and the quotient is $x^4+12x^3+21x^2-62x-72$ .

Concept used:

If the remainder is $0$ the divisor is the factor of the equation.

In remainder theorem the polynomials equation can be divided by eliminating the term by term.

Calculation:

According to the given the divisor is $x+4$ and the quotient is $x^4+12x^3+21x^2-62x-72$ .

Using remainder theorem:

  $x+4\begin{array}{c}\hfill x^3-8x^2-11x-18\\ \hfill \overline{)\begin{array}{l}\underset{\_}{\begin{array}{l}{x}^4+12x^3+21x^2-62x-72\\ x^4+4x^3\end{array}}\\ \underset{\_}{\begin{array}{l}-8x^3+21x^2-62x-72\\ -8x^3-32x^2\end{array}}\\ \underset{\_}{\begin{array}{l}-11x^2-62x-72\\ -11x^2-44x\end{array}}\\ \underset{\_}{\begin{array}{l}-18x-72\\ -18x-72\end{array}}\\ 0\\ \end{array}}\end{array}$

Hence, the remainder is $0$ and the binomial is factor of the polynomial.