Question

divide the expression.\\(\frac{8x^{3}+10x^{2}-12x - 15}{2x^{2}-3}\\)\\(\\)write your answer in the form \\((\text{quotient})+\frac{(\text{remainder})}{(\text{divisor})}\\).\\(\frac{8x^{3}+10x^{2}-12x - 15}{2x^{2}-3}=\square\\)

Explanation:

Step1: Divide the leading terms

Divide the leading term of the numerator \(8x^3\) by the leading term of the denominator \(2x^2\). We get \(\frac{8x^3}{2x^2} = 4x\).

Step2: Multiply the divisor by \(4x\)

Multiply the divisor \(2x^2 - 3\) by \(4x\): \(4x(2x^2 - 3)=8x^3 - 12x\).

Step3: Subtract this from the numerator

Subtract \(8x^3 - 12x\) from the numerator \(8x^3 + 10x^2 - 12x - 15\):
\[

$$\begin{align*} &(8x^3 + 10x^2 - 12x - 15)-(8x^3 - 12x)\\ =&8x^3 + 10x^2 - 12x - 15 - 8x^3 + 12x\\ =&10x^2 - 15 \end{align*}$$

\]

Step4: Divide the new leading term

Now, divide the leading term of the new numerator \(10x^2\) by the leading term of the denominator \(2x^2\). We get \(\frac{10x^2}{2x^2}=5\).

Step5: Multiply the divisor by \(5\)

Multiply the divisor \(2x^2 - 3\) by \(5\): \(5(2x^2 - 3)=10x^2 - 15\).

Step6: Subtract this from the new numerator

Subtract \(10x^2 - 15\) from \(10x^2 - 15\):
\[

$$\begin{align*} &(10x^2 - 15)-(10x^2 - 15)\\ =&10x^2 - 15 - 10x^2 + 15\\ =&0 \end{align*}$$

\]
Since the remainder is \(0\), the division is exact. The quotient is \(4x + 5\) and the remainder is \(0\).

Answer:

\(4x + 5+\frac{0}{2x^2 - 3}\) (or simply \(4x + 5\) since the remainder is \(0\))