Chapter 4, Problem 13RE

To determine

To calculate: The remainder R when $f(x)=x^4-2x^3+15x-2$ is divided by $g(x)=x+2$ .

Also, to find out if g a factor of f .

Answer to Problem 13RE

The remainder is $0$ and g is a factor of f .

Explanation of Solution

Given information:

  $f(x)=x^4-2x^3+15x-2$ and $g(x)=x+2$ .

Formula used:

When $f(x)$ is divided by a simple polynomial say $(x-c)$ we get,

  $f(x)=(x-c).q(x)+r(x)$

Where $r(x)$ is the remainder with either the zero polynomial or a polynomial function of degree 0.

Remainder Theorem:

The theorem states when we divide a polynomial $f(x)$ by $(x-c)$ the remainder is $f(c)$ .

Factor Theorem:

The theorem states $(x-c)$ is a factor of $f(x)$ if and only if $f(c)=0$ .

Calculation:

Consider ,

  $f(x)=x^4-2x^3+15x-2$

And

  $\begin{array}{l}g(x)=x+2\\ g(x)=x-(-2)\end{array}$

Since $g(x)$ is of the form $(x-c)$

  $=c=-2$ .

Now ,

  $\begin{array}{l}f(-2)={(}^{-}-2{(}^{-}+15(-2)-2\\ =16+16-30-2\\ =32-32\\ =0\end{array}$

The remainder is 0.

Also,

  $\begin{array}{l}=f(c)=0\\ =0=0\end{array}$ .

Hence , the remainder is 0 and g is a factor of f .