Question

divide \left(28b^3-14b^2-66b+26\right)\div\left(4b-6\right)

Answer 1

To perform the polynomial long division of \(28b^3 - 14b^2 - 66b + 26\) by \(4b - 6\), we follow these steps:

1. **Divide the leading term:**
- Divide the leading term of the numerator, \(28b^3\), by the leading term of the denominator, \(4b\):
\[
\frac{28b^3}{4b} = 7b^2
\]

2. **Multiply and subtract:**
- Multiply \(7b^2\) by the entire divisor \(4b - 6\):
\[
7b^2(4b - 6) = 28b^3 - 42b^2
\]
- Now subtract this from the original polynomial:
\[
(28b^3 - 14b^2 - 66b + 26) - (28b^3 - 42b^2) = ( -14b^2 + 42b^2) - 66b + 26 = 28b^2 - 66b + 26
\]

3. **Repeat the process:**
- Now divide the new leading term \(28b^2\) by the leading term of the divisor \(4b\):
\[
\frac{28b^2}{4b} = 7b
\]
- Multiply \(7b\) by \(4b - 6\):
\[
7b(4b - 6) = 28b^2 - 42b
\]
- Subtract:
\[
(28b^2 - 66b + 26) - (28b^2 - 42b) = (-66b + 42b) + 26 = -24b + 26
\]

4. **Continue dividing:**
- Now divide \(-24b\) by \(4b\):
\[
\frac{-24b}{4b} = -6
\]
- Multiply \(-6\) by \(4b - 6\):
\[
-6(4b - 6) = -24b + 36
\]
- Subtract:
\[
(-24b + 26) - (-24b + 36) = 26 - 36 = -10
\]

5. **Final result:**
- The result of the division is:
\[
7b^2 + 7b - 6 + \frac{-10}{4b - 6}
\]

So, the final answer is:
\[
7b^2 + 7b - 6 - \frac{10}{4b - 6}
\]