Answered

Consider this quotient.

$\left(x^3-8 x+6

ight) \div\left(x^2-2 x+1

ight)$



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.



Options:

$x^2-2 x+1$

$x^3-8 x+6$

$-11 x+12$

$x+2$

$-5 x+4$

$x-6$

Verified Answer :

  • Divide the leading term of the dividend by the divisor to find the first term of the quotient.
  • Multiply and subtract to find the new partial remainder, then repeat the process until the remainder has a degree lower than the divisor.
  • The resulting division gives the quotient \(q(x) = x + 2\) and the remainder \(r(x) = -5x + 4\).
  • The final expression is \(\boxed{{(x + 2) + \frac{-5x + 4}{x^2 - 2x + 1}}}\).

Explanation

  1. Understanding the Problem and Setup

To perform the polynomial division of \(P(x) = x^3 - 8x + 6\) by \(B(x) = x^2 - 2x + 1\), we set up the long division. The dividend is \(x^3 + 0x^2 - 8x + 6\) and the divisor is \(x^2 - 2x + 1\). We look for a quotient \(q(x)\) and a remainder \(r(x)\) such that \(\frac{x^3 - 8x + 6}{x^2 - 2x + 1} = q(x) + \frac{r(x)}{B(x)}\).

  1. Performing Polynomial Long Division

First, we divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x^2\) to get \(x\). Then we multiply the entire divisor by \(x\):
\(x(x^2 - 2x + 1) = x^3 - 2x^2 + x\).
Subtracting this from the dividend:
\((x^3 + 0x^2 - 8x + 6) - (x^3 - 2x^2 + x) = 2x^2 - 9x + 6\).
Next, we divide the leading term of the new polynomial \(2x^2\) by the leading term of the divisor \(x^2\) to get \(2\). Multiplying the divisor by 2:
\(2(x^2 - 2x + 1) = 2x^2 - 4x + 2\).
Subtracting this from our current remainder:
\((2x^2 - 9x + 6) - (2x^2 - 4x + 2) = -5x + 4\).
Since the degree of \(-5x + 4\) is less than the degree of the divisor, the division is complete.

  1. Conclusion and Final Form

The division process yields a quotient \(q(x) = x + 2\) and a remainder \(r(x) = -5x + 4\). The divisor is \(b(x) = x^2 - 2x + 1\). Thus, the original quotient can be written as \((x + 2) + \frac{-5x + 4}{x^2 - 2x + 1}\). Comparing this to the given options, we identify the correct components as \(q(x) = x + 2\), \(r(x) = -5x + 4\), and \(b(x) = x^2 - 2x + 1\).

Examples

Polynomial long division is essential in engineering and physics when simplifying complex transfer functions or breaking down rational expressions into simpler components (like partial fraction decomposition). For example, if a signal processing filter is represented by the ratio of two polynomials, performing division allows engineers to isolate the stable part of the filter from the residue, which is vital for analyzing circuit stability and response times. Understanding this method helps in simplifying complex mathematical models into manageable, interpretable forms.

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