Chapter 5, Problem 19SGR

To determine

To calculate: The expression,

  $\left(a^4+5a^3+2a^2-6a+4\right){\left(a+2\right)}^{-1}$

Answer to Problem 19SGR

The factor of solution of the expression is $\left(a+2\right)\left(a^3+3a^2-4a+2\right)$ .

Explanation of Solution

Given information:

The expression is given as:

  $\left(a^4+5a^3+2a^2-6a+4\right){\left(a+2\right)}^{-1}$

Formula used:

The process when divide a polynomial by a polynomial with more than one term is known as division algorithm.

Calculation:

Consider the expression,

  $\left(a^4+5a^3+2a^2-6a+4\right){\left(a+2\right)}^{-1}$

The expression can be written as,

  $\frac{\left(a^4+5a^3+2a^2-6a+4\right)}{\left(a+2\right)}$

Using the long division method,

  $a+2\overline{)a^4+5a^3+2a^2-6a+4}$

Multiply divisor by $a^3$ since $\frac{a^4}{a^3}=a$ .

Subtract and bring down the next term as,

  $a+2\begin{array}{c}\hfill a^3\\ \hfill \overline{)\begin{array}{l}{a}^4+5a^3+2a^2-6a+4\\ a^4+2a^3\\ --\\ \overline{\text{}3a^3}\end{array}}\end{array}$

Now, multiply divisor by $3a^2$ since, $\frac{3a^3}{3a^2}=a$ .

Subtract and bring down the next term as,

  $a+2\begin{array}{c}\hfill a^3+3a^2\\ \hfill \overline{)\begin{array}{l}{a}^4+5a^3+2a^2-6a+4\\ a^4+2a^3\\ --\\ \overline{\begin{array}{l}\text{}3a^3+2a^2-6a+4\\ 3a^3+6a^2\\ --\\ \overline{-4a^2-6a+4}\end{array}}\end{array}}\end{array}$

Now, multiply divisor by $-4a$ since, $\frac{-4a^2}{-4a}=a$ .

Subtract and bring down the next term as,

  $a+2\begin{array}{c}\hfill a^3+3a^2-4a\\ \hfill \overline{)\begin{array}{l}{a}^4+5a^3+2a^2-6a+4\\ a^4+2a^3\\ --\\ \overline{\begin{array}{l}\text{}3a^3+2a^2-6a+4\\ 3a^3+6a^2\\ --\\ \overline{\begin{array}{l}-4a^2-6a+4\\ -4a^2-8a\\ ++\\ \overline{2a+4}\end{array}}\end{array}}\end{array}}\end{array}$

Now, multiply divisor by $2$ since, $\frac{2a}{2}=a$ .

Subtract and bring down the next term as,

The quotient is $a^3+3a^2-4a+2$ and remainder is 0.

Therefore, the factor is $\left(a+2\right)\left(a^3+3a^2-4a+2\right)$ .

Thus, the factor of solution of the expression is $\left(a+2\right)\left(a^3+3a^2-4a+2\right)$