Chapter 3, Problem 23CLT

To determine

To divide: $3x^4+2x^2-5x+3$ by $x-2$ .

Answer to Problem 23CLT

  $3x^3+6x^2+14x+23+\frac{49}{x-2}$

Explanation of Solution

Concept Used:

Synthetic Division (for a Quadratic Polynomial)

To divide $a{x}^2+bx+x$ by $x-k$ , the pattern is:

  

For higher order polynomial similar process is extended.

Calculation: The question asks to divide $3x^4+2x^2-5x+3$ by divisor $x-2$ using synthetic division.

Here $k=2$ .

The given dividend $f\left(x\right)=3x^4+2x^2-5x+3$ can also be written as:

  $f\left(x\right)=3x^4+0x^3+2x^2-5x+3$

Then, use synthetic division to divide $f\left(x\right)=3x^4+0x^3+2x^2-5x+3$ by $x-2$ .

  $\begin{array}{l}\text{ 3 0 2 }-5\text{ 3}\\ \text{ 6 12 28 46}\\ 2\overline{)\text{3 6 14 23 49}}=R\end{array}$

Thus, the quotient is $Q\left(x\right)=3x^3+6x^2+14x+23$ and the remainder is $R\left(x\right)=49$ .

So,

  $\frac{3x^4+2x^2-5x+3}{x-2}=3x^3+6x^2+14x+23+\frac{49}{x-2}$