Certainly! Let's perform the polynomial division of \(8x^3 + 6x^2 + 10x - 1\) by \(1 - 4x\) step by step.
### Step-by-Step Polynomial Division
1. Setup the division:
We are dividing \(8x^3 + 6x^2 + 10x - 1\) by \(1 - 4x\). We'll write our polynomials in the standard form.
2. Divide the leading terms:
The leading term of the dividend \(8x^3\) divided by the leading term of the divisor \(-4x\):
[tex]\[
\frac{8x^3}{-4x} = -2x^2
\][/tex]
This becomes the first term of our quotient.
3. Multiply and subtract:
Multiply the entire divisor \(1 - 4x\) by \(-2x^2\) and subtract it from the original polynomial:
[tex]\[
(1 - 4x) \cdot (-2x^2) = -2x^2 + 8x^3
\][/tex]
Subtract this from \(8x^3 + 6x^2 + 10x - 1\):
[tex]\[
\begin{array}{r}
8x^3 + 6x^2 + 10x - 1 \\
- (8x^3 - 8x^3 + 16x^2) \\
\hline
6x^2 - 16x^2 + 10x - 1 \\
6x^2 - 16x^2 = -10x^2 \\
10x - 1
\end{array}
\][/tex]
So, now, our polynomial becomes \(-10x^2 + 10x - 1\).
4. Repeat the process:
Divide the new leading term \(-10x^2\) by the leading term of the divisor \(-4x\):
[tex]\[
\frac{-10x^2}{-4x} = 2.5x
\][/tex]
Multiply the entire divisor \(1 - 4x\) by \(2.5x\) and subtract:
[tex]\[
(1 - 4x) \cdot (2.5x) = 2.5x - 10x^2
\][/tex]
Subtract this from \(-10x^2 + 10x - 1\):
[tex]\[
\begin{array}{r}
-10x^2 + 10x - 1 \\
- (-10x^2 + 10x) \\
\hline
10x - 10x = 0x - 1 + 10 \\
-1
\end{array}
\][/tex]
Now, our polynomial is reduced to -1.
5. Check the remainder:
Since our polynomial \(-1\) is a constant and cannot be divided further by \(1 - 4x\), this is our remainder.
### Final Answer:
The quotient is:
[tex]\[
-2x^2 + 2.5x
\][/tex]
The remainder is:
[tex]\[
-1
\][/tex]
So, when we divide [tex]\(8x^3 + 6x^2 + 10x - 1\)[/tex] by [tex]\(1 - 4x\)[/tex], the quotient is [tex]\(-2x^2 + 2.5x\)[/tex] and the remainder is [tex]\(-1\)[/tex].
