To solve the division [tex]\( \left(x^3 - 8x + 6\right) \div \left(x^2 - 2x + 1\right) \)[/tex] using polynomial long division, let's go through the detailed steps to find the quotient [tex]\( q(x) \)[/tex], the remainder [tex]\( r(x) \)[/tex], and understand the divisor [tex]\( b(x) \)[/tex].
### Step 1: Initialize
We have:
- Dividend (numerator): [tex]\( x^3 - 8x + 6 \)[/tex]
- Divisor (denominator): [tex]\( x^2 - 2x + 1 \)[/tex]
### Step 2: Perform the Division
1. First Division Step:
- Divide the leading term of the dividend by the leading term of the divisor:
[tex]\[
\frac{x^3}{x^2} = x
\][/tex]
- Multiply the whole divisor by [tex]\( x \)[/tex]:
[tex]\[
(x^2 - 2x + 1) \cdot x = x^3 - 2x^2 + x
\][/tex]
- Subtract this from the dividend:
[tex]\[
(x^3 - 8x + 6) - (x^3 - 2x^2 + x) = 2x^2 - 9x + 6
\][/tex]
2. Second Division Step:
- Divide the new leading term of the result by the leading term of the divisor:
[tex]\[
\frac{2x^2}{x^2} = 2
\][/tex]
- Multiply the whole divisor by [tex]\( 2 \)[/tex]:
[tex]\[
(x^2 - 2x + 1) \cdot 2 = 2x^2 - 4x + 2
\][/tex]
- Subtract this from the current result:
[tex]\[
(2x^2 - 9x + 6) - (2x^2 - 4x + 2) = -5x + 4
\][/tex]
- We see that the remainder is now [tex]\(-5x + 4\)[/tex], which has a degree less than the divisor [tex]\((x^2 - 2x + 1)\)[/tex].
### Step 3: Conclusion
From this process, we obtain:
- Quotient [tex]\( q(x) = x + 2 \)[/tex]
- Remainder [tex]\( r(x) = -5x + 4 \)[/tex]
- Divisor [tex]\( b(x) = x^2 - 2x + 1 \)[/tex]
Therefore, we can rewrite the original quotient in the form of [tex]\( q(x) + \frac{r(x)}{b(x)} \)[/tex] as:
[tex]\[
x + 2 + \frac{-5x + 4}{x^2 - 2x + 1}
\][/tex]
So, the correct expressions to be placed are:
- [tex]\( q(x) = x + 2 \)[/tex]
- [tex]\( r(x) = -5x + 4 \)[/tex]
- [tex]\( b(x) = x^2 - 2x + 1 \)[/tex]
