Questions: Consider this quotient. [ left(x^3-8 x+6right) divleft(x^2-2 x+1right) ] Use long division to rewrite the quotient in an equivalent form as (q(x)+fracr(x)b(x)), where (q(x)) is the quotient, (r(x)) is the remainder, and (b(x)) is the divisor. [ x+2 quad-5 x+4 quad-11 x+12 quad x^2-2 x+1 quad x-6 quad x^3-8 x+6 ]

$Consider this quotient. [ left(x^3-8 x+6right) divleft(x^2-2 x+1right) ] Use long division to rewrite the quotient in an equivalent form as (q(x)+fracr(x)b(x)), where (q(x)) is the quotient, (r(x)) is the remainder, and (b(x)) is the divisor. [ x+2 quad-5 x+4 quad-11 x+12 quad x^2-2 x+1 quad x-6 quad x^3-8 x+6 ]$

Transcript text: Consider this quotient. \[ \left(x^{3}-8 x+6\right) \div\left(x^{2}-2 x+1\right) \] Use long division to rewrite the quotient in an equivalent form as $q(x)+\frac{r(x)}{b(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor. \[ x+2 \quad-5 x+4 \quad-11 x+12 \quad x^{2}-2 x+1 \quad x-6 \quad x^{3}-8 x+6 \]

Solution

Solution Steps

To solve the problem of dividing the polynomial $x^3 - 8x + 6$ by $x^2 - 2x + 1$ using polynomial long division, follow these steps:

Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x^2$ to get the first term of the quotient.
Multiply the entire divisor by this term and subtract the result from the dividend.
Repeat the process with the new polynomial obtained after subtraction until the degree of the remainder is less than the degree of the divisor.
The quotient obtained is $q(x)$ and the remainder is $r(x)$.

Step 1: Set Up the Polynomial Division

To divide the polynomial $x^3 - 8x + 6$ by $x^2 - 2x + 1$, we perform polynomial long division. The goal is to express the division in the form $q(x) + \frac{r(x)}{b(x)}$, where $q(x)$ is the quotient, $r(x)$ is the remainder, and $b(x)$ is the divisor.

Step 2: Perform the Division

Divide the leading term of the dividend $x^3$ by the leading term of the divisor $x^2$ to get the first term of the quotient: $x$.
Multiply the entire divisor $x^2 - 2x + 1$ by $x$ to get $x^3 - 2x^2 + x$.
Subtract this result from the original dividend $x^3 - 8x + 6$ to get a new polynomial: $2x^2 - 9x + 6$.
Repeat the process with the new polynomial:
- Divide $2x^2$ by $x^2$ to get $2$.
- Multiply the divisor by $2$ to get $2x^2 - 4x + 2$.
- Subtract this from $2x^2 - 9x + 6$ to get the remainder $-5x + 4$.

Step 3: Express the Result

The quotient $q(x)$ is $x + 2$ and the remainder $r(x)$ is $-5x + 4$. Therefore, the division can be expressed as:

\[ x^3 - 8x + 6 = (x^2 - 2x + 1)(x + 2) + (-5x + 4) \]

This can be rewritten in the form:

\[ \frac{x^3 - 8x + 6}{x^2 - 2x + 1} = x + 2 + \frac{-5x + 4}{x^2 - 2x + 1} \]