Chapter 6.8, Problem 12STP

To determine

To calculate: The expression,

  $\frac{3x^4-4x^2-28x-16}{x+2}$

Answer to Problem 12STP

The factor of solution of the expression is $3x^3-6x^2+8x-44+\frac{72}{x+2}$ .

Explanation of Solution

Given information:

The expression is given as:

  $\frac{3x^4-4x^2-28x-16}{x+2}$

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,

  $\frac{3x^4-4x^2-28x-16}{x+2}$

Using the long division method,

  $x+2\overline{)3x^4-4x^2-28x-16}$

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

Subtract and bring down the next term as,

  $\begin{array}{l}x+2\begin{array}{c}\hfill 3x^3\\ \hfill \overline{)\begin{array}{l}3x^4-4x^2-28x-16\\ 3x^4+6x^3\\ --\\ \overline{6x^3-4x^2-28x-16}\end{array}}\end{array}\end{array}$

Now, multiply divisor by $6x^2$ since, $\frac{6x^3}{6x^2}=x$ .

Subtract and bring down the next term as,

  $\begin{array}{l}x+2\begin{array}{c}\hfill 3x^3-6x^2\\ \hfill \overline{)\begin{array}{l}3x^4-4x^2-28x-16\\ 3x^4+6x^3\\ --\\ \overline{\begin{array}{l}-6x^3-4x^2-28x-16\\ -6x^3-12x^2\\ ++\\ \overline{8x^2-28x-16}\end{array}}\end{array}}\end{array}\end{array}$

Now, multiply divisor by $8x$ since, $\frac{8x^2}{8x}=x$ .

Subtract and bring down the next term as,

  $x+2\begin{array}{c}\hfill 3x^3-6x^2+8x\\ \hfill \overline{)\begin{array}{l}3x^4-4x^2-28x-16\\ 3x^4+6x^3\\ --\\ \overline{\begin{array}{l}-6x^3-4x^2-28x-16\\ -6x^3-12x^2\\ --\\ \overline{\begin{array}{l}8x^2-28x-16\\ 8x^2+16x\\ --\\ \overline{-44x-16}\end{array}}\end{array}}\end{array}}\end{array}$

Now, multiply divisor by $-44$ since, $\frac{-44x}{-44}=x$ .

Subtract and bring down the next term as,

  $x+2\begin{array}{c}\hfill 3x^3-6x^2+8x-44\\ \hfill \overline{)\begin{array}{l}3x^4-4x^2-28x-16\\ 3x^4+6x^3\\ --\\ \overline{\begin{array}{l}-6x^3-4x^2-28x-16\\ -6x^3-12x^2\\ --\\ \overline{\begin{array}{l}8x^2-28x-16\\ 8x^2+16x\\ --\\ \overline{\begin{array}{l}-44x-16\\ -44x-88\\ ++\\ \overline{72}\end{array}}\end{array}}\end{array}}\end{array}}\end{array}$

The quotient is $3x^3-6x^2+8x-44$ and remainder is 72.

Therefore, the factor is $3x^3-6x^2+8x-44+\frac{72}{x+2}$ .

Thus, the factor of solution of the expression is $3x^3-6x^2+8x-44+\frac{72}{x+2}$