### Instruction: What is the quotient when $8x^3+16x^2-7x+4$ is divided by $2x+5$?### Response: I want to find a polynomial $Q(x)$ such that $8x^3+16x^2-7x+4 = (2x+5)Q(x)$.
I can use long division to find $Q(x)$, just like I would do with numbers.
I start by asking myself, what do I need to multiply $2x$ by to get $8x^3$?
The answer is $4x^2$.
So I write $4x^2$ as the first term of $Q(x)$, and I multiply it by $2x+5$ to get $8x^3+20x^2$.
I subtract this from the original polynomial to get the remainder $-4x^2-7x+4$.
I repeat the process with the new remainder.
What do I need to multiply $2x$ by to get $-4x^2$?
The answer is $-2x$.
So I write $-2x$ as the second term of $Q(x)$, and I multiply it by $2x+5$ to get $-4x^2-10x$.
I subtract this from the previous remainder to get the new remainder $3x+4$.
I do it one more time.
What do I need to multiply $2x$ by to get $3x$?
The answer is $\frac{3}{2}$.
So I write $\frac{3}{2}$ as the third term of $Q(x)$, and I multiply it by $2x+5$ to get $3x+\frac{15}{2}$.
I subtract this from the last remainder to get the final remainder $\frac{-7}{2}$.
I cannot divide $2x+5$ into this any further, so I stop here.
The final quotient is $Q(x) = 4x^2-2x+\frac{3}{2}$, and the final remainder is $\frac{-7}{2}$.
I can write the answer as $Q(x) + \frac{R(x)}{2x+5}$, where $R(x)$ is the remainder polynomial.
In this case, $R(x) = -7$, so the answer is $4x^2-2x+\frac{3}{2} - \frac{7}{2x+5}$.