Question

Let f(x) = 8x3 − 22x2 − 4 and g(x) = 4x − 3. Find f of x over g of x.

2x2 − 4x − 3 − 13 over quantity 4x minus 3
2x2 − 4x − 3 − quantity of 4 x minus 3 over 13
2x2 − 7x − 1
2x2 − 7x − 5 + quantity of x minus 4 over quantity of 4 x minus 3

Answer 1

To find \( \frac{f(x)}{g(x)} \), we first calculate the values of \( f(x) \) and \( g(x) \).

Given:
\[
f(x) = 8x^3 - 22x^2 - 4
\]
\[
g(x) = 4x - 3
\]

Now we need to compute \( \frac{f(x)}{g(x)} \):
\[
\frac{f(x)}{g(x)} = \frac{8x^3 - 22x^2 - 4}{4x - 3}
\]

To simplify this expression, we can perform polynomial long division.

1. Divide the leading term \( 8x^3 \) by \( 4x \), which gives \( 2x^2 \).
2. Multiply \( 2x^2 \) by \( g(x) \):
\[
2x^2(4x - 3) = 8x^3 - 6x^2
\]
3. Subtract this from \( f(x) \):
\[
f(x) - (8x^3 - 6x^2) = (8x^3 - 22x^2 - 4) - (8x^3 - 6x^2) = -16x^2 - 4
\]
4. Now take the new polynomial \(-16x^2 - 4\) and divide the leading term \(-16x^2\) by \(4x\):
\[
-16x^2 / 4x = -4x
\]
5. Multiply \(-4x\) by \(g(x)\):
\[
-4x(4x - 3) = -16x^2 + 12x
\]
6. Subtract this from the current result:
\[
-16x^2 - 4 - (-16x^2 + 12x) = -4 - 12x = -12x - 4
\]
7. Now, divide \(-12x\) by \(4x\):
\[
-12x / 4x = -3
\]
8. Multiply \(-3\) by \(g(x)\):
\[
-3(4x - 3) = -12x + 9
\]
9. Finally, subtract this from the current result:
\[
-12x - 4 - (-12x + 9) = -4 - 9 = -13
\]

Thus, putting everything together, we find:
\[
\frac{f(x)}{g(x)} = 2x^2 - 4x - 3 - \frac{13}{4x - 3}
\]

Now, reviewing the options you provided, we can see that the first option fits this format:
\[
2x^2 - 4x - 3 - \frac{13}{4x - 3}
\]

Therefore, the answer is:
\[
\boxed{2x^2 - 4x - 3 - \frac{13}{4x - 3}}
\]