Question

Show that the equation $p\left(x\right)=q\left(x\right)m\left(x\right)+R\left(x\right)$is equivalent to the equation $\frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)}$for all x for which q(x) is nonzero.

Short answer

The given equation has been proved.

Step-by-step solution

Step 1. Given Information

The given equation is$p\left(x\right)=q\left(x\right)m\left(x\right)+R\left(x\right)$

Step 2. Proof

Divide both the sides of the given equation by q(x) after that simplify the equation.

$$\begin{gathered} \frac{p\left(x\right)}{q\left(x\right)}=\frac{q\left(x\right)m\left(x\right)+R\left(x\right)}{q\left(x\right)} \\ \frac{p\left(x\right)}{q\left(x\right)}=\frac{q\left(x\right)m\left(x\right)}{q\left(x\right)}+\frac{R\left(x\right)}{q\left(x\right)} \\ \frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)} \end{gathered}$$

Thus, the equation is proved.