Chapter 4.3, Problem 22E

To determine

To match: the equivalent expression.

Answer to Problem 22E

The answer is option B.

Explanation of Solution

Given information:

Given expression,

  $\left(x^2+x+3\right)÷\left(x-2\right)$

Given options are,

  $x+1-\frac{1}{x-2}$

  $x+3+\frac{9}{x-2}$

  $x+1+\frac{5}{x-2}$

  $x+3+\frac{3}{x-2}$

Calculation:

Write the coefficient of the dividend in order of descending exponents. Use 0 for the missing $x$ term.

Because the divisor is $x-2$ , so $k=2$ . Write 2 to the left vertical bar.

Bring down the leading coefficient. Multiply the leading coefficient by the 2. Write the product under the second coefficient then add.

Multiply the previous sum 3 by 2 . Write the product under the third coefficient then add. Repeat this process for the remaining coefficient.

  $\begin{array}{cccc}2& 1& 1& 3\\ & & 2& 6\\ & 1& 1& 9\end{array}$

The quotient is $x\left(1\right)+3=x+3$ and remainder is 9.

Therefore, the result is $x+3+\frac{9}{x-2}$ .

So, the answer is option B.