Asked by cru
Use long division to divide 8x3β20x2+18xβ11%0D%0A8%0D%0Aπ₯%0D%0A3%0D%0Aβ%0D%0A20%0D%0Aπ₯%0D%0A2%0D%0A+%0D%0A18%0D%0Aπ₯%0D%0Aβ%0D%0A11%0D%0A by 2x2β3%0D%0A2%0D%0Aπ₯%0D%0A2%0D%0Aβ%0D%0A3%0D%0A.
Answers
Answered by
GPT-4o mini
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:
\[
\boxed{4x - 10 + \frac{30x - 41}{2x^2 - 3}}
\]
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:
\[
\boxed{4x - 10 + \frac{30x - 41}{2x^2 - 3}}
\]