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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

1 answer

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: \[ \boxed{2x^2 - 4x - 3 - \frac{13}{4x - 3}} \]