Chapter 7.9, Problem 9MRE

To determine

To calculate: The simplified form of the expression $\frac{a^3+27}{a-3}$ .

Answer to Problem 9MRE

The simplified form of the expression $\frac{a^3+27}{a-3}$ is $\left(a^2+3a+9\right)+\frac{54}{\left(a-3\right)}$ .

Explanation of Solution

Given information:

The expression $\frac{a^3+27}{a-3}$ .

Formula used:

When a polynomial is divided by its factor then dividend is the product of divisor and quotient increased by remainder.

Calculation:

Consider the expression $\frac{a^3+27}{a-3}$ .

When a polynomial is divided by its factor then dividend is the product of divisor and quotient increased by remainder.

Apply the method of long division. First multiply the expression $a-3$ by $a^2$ . Subtract it from the dividend. Next multiply $a-3$ by $3a$ . Subtract the expression obtained.

  $\begin{array}{l}a-3\begin{array}{c}\hfill a^2+3a+9\\ \hfill \overline{)a^3+27}\end{array}\\ \begin{array}{ccc}& & a^3-3a^2\end{array}\\ \begin{array}{ccc}& & \begin{array}{cc}& 3a^2+27\end{array}\end{array}\\ \begin{array}{cc}& \begin{array}{ccc}& & 3a^2-9a\end{array}\end{array}\\ \begin{array}{cccccc}& & & & 9a+27& \end{array}\\ \begin{array}{cccccc}& & & & 9a-27& \end{array}\\ \begin{array}{cccccc}& & & & & 54\end{array}\end{array}$

Now, since $\text{Dividend}=\text{Quotient}\times \text{Divisor}+\text{Remainder}$ . Here, dividend is $a^3+27$ quotient is $a^2+3a+9$ , divisor is $a-3$ and remainder is $54$ .

Therefore, $a^3+27=\left(a-3\right)\left(a^2+3a+9\right)+54$ that is,

  $\frac{a^3+27}{a-3}=\left(a^2+3a+9\right)+\frac{54}{\left(a-3\right)}$

Thus, the simplified form of the expression $\frac{a^3+27}{a-3}$ is $\left(a^2+3a+9\right)+\frac{54}{\left(a-3\right)}$ .