Question

using long division, what is the quotient of this expression?\\(\frac{x^{4}+4x^{3}-5x^{2}+x - 3}{x^{2}+x - 3}\\)\\(\text{a. } x^{2}+18x - 23\\)\\(\text{b. } x^{2}+3x - 5+\frac{15x - 18}{x^{2}+x - 3}\\)\\(\text{c. } x^{2}+3x - 5\\)\\(\text{d. } x^{2}+3x - 5-\frac{15x - 18}{x^{2}+x - 3}\\)

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(x^4 + 4x^3 - 5x^2 + x - 3\) (\(x^4\)) by the leading term of the divisor \(x^2 + x - 3\) (\(x^2\)) to get the first term of the quotient: \(\frac{x^4}{x^2}=x^2\).

Step2: Multiply and subtract

Multiply the divisor \(x^2 + x - 3\) by \(x^2\) to get \(x^4 + x^3 - 3x^2\). Subtract this from the dividend:
\[

$$\begin{align*} &(x^4 + 4x^3 - 5x^2 + x - 3)-(x^4 + x^3 - 3x^2)\\ =&x^4 + 4x^3 - 5x^2 + x - 3 - x^4 - x^3 + 3x^2\\ =&3x^3 - 2x^2 + x - 3 \end{align*}$$

\]

Step3: Divide the new leading terms

Divide the leading term of the new dividend \(3x^3\) by the leading term of the divisor \(x^2\) to get the next term of the quotient: \(\frac{3x^3}{x^2}=3x\).

Step4: Multiply and subtract again

Multiply the divisor \(x^2 + x - 3\) by \(3x\) to get \(3x^3 + 3x^2 - 9x\). Subtract this from the new dividend:
\[

$$\begin{align*} &(3x^3 - 2x^2 + x - 3)-(3x^3 + 3x^2 - 9x)\\ =&3x^3 - 2x^2 + x - 3 - 3x^3 - 3x^2 + 9x\\ =& - 5x^2 + 10x - 3 \end{align*}$$

\]

Step5: Divide the new leading terms again

Divide the leading term of the new dividend \(-5x^2\) by the leading term of the divisor \(x^2\) to get the next term of the quotient: \(\frac{-5x^2}{x^2}=-5\).

Step6: Multiply and subtract the last time

Multiply the divisor \(x^2 + x - 3\) by \(-5\) to get \(-5x^2 - 5x + 15\). Subtract this from the new dividend:
\[

$$\begin{align*} &(-5x^2 + 10x - 3)-(-5x^2 - 5x + 15)\\ =&-5x^2 + 10x - 3 + 5x^2 + 5x - 15\\ =&15x - 18 \end{align*}$$

\]
The quotient is \(x^2 + 3x - 5\) and the remainder is \(15x - 18\). But when we are asked for the quotient (not including the remainder part in the quotient), the quotient is \(x^2 + 3x - 5\).

Answer:

C. \(x^2 + 3x - 5\)