Question

24
enter the correct answer in the box.
use long division to determine the quotient of the following expression.
$(10x^2 + 13x - 77) \div (2x + 7)$
write the quotient in standard form with the term of largest degree on the left.

Explanation:

Step1: Divide the leading terms

Divide the leading term of the dividend \(10x^2\) by the leading term of the divisor \(2x\). So, \(\frac{10x^2}{2x} = 5x\). This is the first term of the quotient.

Step2: Multiply and subtract

Multiply the divisor \(2x + 7\) by \(5x\), we get \(5x(2x + 7)=10x^2 + 35x\). Now subtract this from the dividend \(10x^2 + 13x - 77\):
\[

$$\begin{align*} (10x^2 + 13x - 77)-(10x^2 + 35x)&=10x^2 + 13x - 77 - 10x^2 - 35x\\ &=(10x^2-10x^2)+(13x - 35x)-77\\ &=-22x - 77 \end{align*}$$

\]

Step3: Divide the new leading term

Now, divide the leading term of the new dividend \(-22x\) by the leading term of the divisor \(2x\). So, \(\frac{-22x}{2x}=-11\). This is the second term of the quotient.

Step4: Multiply and subtract (final check)

Multiply the divisor \(2x + 7\) by \(-11\), we get \(-11(2x + 7)=-22x - 77\). Subtract this from the new dividend \(-22x - 77\):
\[
(-22x - 77)-(-22x - 77)=-22x - 77 + 22x + 77 = 0
\]
Since the remainder is 0, the division is exact.

Answer:

\(5x - 11\)