Question

In Exercises 9–11, suppose that you apply polynomial long division to divide a polynomial p(x) by a polynomial q(x) with the goal of obtaining an expression of the form $\frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)}$

If the degree of p(x) is equal to the degree of q(x), what can you say about the degrees of m(x) and R(x)?

Short answer

The degree of m(x) is 0 and degree of R(x) is strictly less than a degree of q(x).

Step-by-step solution

Step 1. Given Information 

The given data is thatIf the degree of p(x) is equal to the degree of q(x).

Step 2. Calculation 

For some polynomial m(x) of degree $n-m$and some proper rational function $\frac{R\left(x\right)}{q\left(x\right)}\mathrm{with}deg\left(R\right(x\left)\right)<m$.

It is given that$n=m$

So, according to above definition,

$$\begin{gathered} m\left(x\right)=n-m \\ =m-m \\ =0 \end{gathered}$$

Hence, degree of m(x) is 0 and degree of R(x) is strictly less than a degree ofq(x).