Use long division to divide 8x3−20x2+18x−11%0D%0A8%0D%0A𝑥%0D%0A3%0D%0A−%0D%0A20%0D%0A𝑥%0D%0A2%0D%0A+%0D%0A18%0D%0A𝑥%0D%0A−%0D%0A11%0D%0A by 2x2−3%0D%0A2%0D%0A𝑥%0D%0A2%0D%0A−%0D%0A3%0D%0A.

Answer 1

To divide the polynomial \(8x^3 - 20x^2 + 18x - 11\) by \(2x^2 - 3\) using long division, we follow these steps:

1. Setup: Write the division in long division format.

   \[ \begin{array}{r|r} 2x^2 - 3 & 8x^3 - 20x^2 + 18x - 11 \ \end{array} \]

2. Divide the leading terms: Divide the leading term of the numerator by the leading term of the denominator.

   \[ \frac{8x^3}{2x^2} = 4x \]

3. Multiply: Multiply the entire divisor \(2x^2 - 3\) by \(4x\).

   \[ 4x \cdot (2x^2 - 3) = 8x^3 - 12x \]

4. Subtract: Subtract this result from the original polynomial.

   \[ (8x^3 - 20x^2 + 18x - 11) - (8x^3 - 12x) = -20x^2 + 18x + 12x - 11 = -20x^2 + 30x - 11 \]

5. Repeat: Now, repeat the process with the new polynomial \(-20x^2 + 30x - 11\):

   • Divide the leading term:

   \[ \frac{-20x^2}{2x^2} = -10 \]

   • Multiply:

   \[ -10 \cdot (2x^2 - 3) = -20x^2 + 30 \]

   • Subtract:

   \[ (-20x^2 + 30x - 11) - (-20x^2 + 30) = 30x - 11 - 30 = 30x - 41 \]

6. Final result: We now have a remainder of \(30x - 41\), which has a lower degree than the divisor \(2x^2 - 3\).

Thus, the final result of the division is:

\[ 4x - 10 + \frac{30x - 41}{2x^2 - 3} \]

So, the complete answer is:

\[ \boxed{4x - 10 + \frac{30x - 41}{2x^2 - 3}} \]