Answer:
Explanation:
To perform long division with the polynomial \(8x^4 - 24x^3 + 16x^2 - 31x + \frac{49}{2}\) divided by \(2x - 5\), follow these steps:
1. Divide the first term of the dividend (\(8x^4\)) by the first term of the divisor (\(2x\)). This gives you \(4x^3\) as the first term of the quotient.
2. Multiply the entire divisor (\(2x - 5\)) by the first term of the quotient (\(4x^3\)). This gives you \(4x^3(2x - 5) = 8x^4 - 20x^3\).
3. Subtract this product from the dividend:
\[
\begin{align*}
&(8x^4 - 24x^3 + 16x^2 - 31x + \frac{49}{2}) - (8x^4 - 20x^3)\\
&= -24x^3 + 16x^2 - 31x + \frac{49}{2}
\end{align*}
\]
4. Bring down the next term of the dividend, which is \(16x^2\), to get \(-24x^3 + 16x^2\).
5. Divide the first term of this new expression (\(-24x^3\)) by the first term of the divisor (\(2x\)), giving you \(-12x^2\) as the next term of the quotient.
6. Multiply the divisor (\(2x - 5\)) by the new term of the quotient (\(-12x^2\)), giving you \(-12x^2(2x - 5) = -24x^3 + 60x^2\).
7. Subtract this product from the new expression:
\[
\begin{align*}
&(-24x^3 + 16x^2) - (-24x^3 + 60x^2)\\
&= 16x^2 - 60x^2\\
&= -44x^2
\end{align*}
\]
8. Bring down the next term of the dividend, which is \(-31x\), to get \(-44x^2 - 31x\).
9. Divide the first term of this new expression (\(-44x^2\)) by the first term of the divisor (\(2x\)), giving you \(-22x\) as the next term of the quotient.
10. Multiply the divisor (\(2x - 5\)) by the new term of the quotient (\(-22x\)), giving you \(-22x(2x - 5) = -44x^2 + 110x\).
11. Subtract this product from the new expression:
\[
\begin{align*}
&(-44x^2 - 31x) - (-44x^2 + 110x)\\
&= -31x - (-31x)\\
&= 0
\end{align*}
\]
12. There is no remainder left, and you have successfully divided the polynomial. Your quotient is \(4x^3 - 12x^2 - 22x\), and there is no remainder.
So, \(8x^4 - 24x^3 + 16x^2 - 31x + \frac{49}{2}\) divided by \(2x - 5\) equals \(4x^3 - 12x^2 - 22x\).