Question

Fill in the blanks to complete each of the following theorem statements:

By applying polynomial long division to divide a polynomial $p\left(x\right)$by a polynomial $q\left(x\right)$of equal or lower degree, we can obtain polynomials $m\left(x\right)$and $R\left(x\right)$such that the following equation is true:

$p\left(x\right)=\_\_\_\_\_+\_\_\_\_\_.$Equivalently, we have the following equation: $\frac{p\left(x\right)}{q\left(x\right)}=\_\_\_\_\_+\_\_\_\_\_.$

Short answer

By applying polynomial long division to divide a polynomial $p\left(x\right)$by a polynomial $q\left(x\right)$of equal or lower degree, we can obtain polynomials $m\left(x\right)$and $R\left(x\right)$such that the following equation is true:

$p\left(x\right)=q\left(x\right)m\left(x\right)+R\left(x\right)$. Equivalently, we have the following equation: $\frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)}$.

Step-by-step solution

Step 1. Given information  

By applying polynomial long division to divide a polynomial $p\left(x\right)$by a polynomial $q\left(x\right)$of equal or lower degree, we can obtain polynomials $m\left(x\right)$and $R\left(x\right)$such that the following equation is true:

$p\left(x\right)=\_\_\_\_\_+\_\_\_\_\_.$Equivalently, we have the following equation: $\frac{p\left(x\right)}{q\left(x\right)}=\_\_\_\_\_+\_\_\_\_\_.$

Step 2. Filling in the blanks to complete the theorem statements  

By applying polynomial long division to divide a polynomial $p\left(x\right)$by a polynomial $q\left(x\right)$of equal or lower degree, we can obtain polynomials $m\left(x\right)$and $R\left(x\right)$such that the following equation is true:

$p\left(x\right)=q\left(x\right)m\left(x\right)+R\left(x\right)$. Equivalently, we have the following equation: $\frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)}$.

The polynomial long division of $p\left(x\right)$divided by $q\left(x\right)$terminates with remainder $R\left(x\right)$whose degree is strictly less than $q\left(x\right)$. The result is sum of polynomial and proper rational function. Where,$m\left(x\right)$is the quotient and$R\left(x\right)$is the remainder of the division process.