Answered

Randy divides $\left(2 x^4-3 x^3-3 x^2+7 x-3\right)$ by $\left(x^2-2 x+1\right)$ as shown below. What error does Randy make?



$\begin{array}{r}

x ^ { 2 } - 2 x + 1 \longdiv { 2 x ^ { 4 } - 3 x ^ { 3 } - 3 x ^ { 2 } + 7 x - 3 } \\

\frac{2 x^4-4 x^3+2 x^2}{x^3-5 x^2+7 x} \\

\frac{x^3-2 x^2+x}{-3 x^2+6 x-3} \\

\frac{-3 x^2+6 x-3}{0}

\end{array}$


AHe makes a subtraction error.
BHe makes an error writing the constant term in the quotient.
CHe makes an error choosing the $x$-term in the quotient.
DHe makes an error rewriting the problem in long division.

Verified Answer :

  • Correctly perform the polynomial long division \((2x^4 - 3x^3 - 3x^2 + 7x - 3) \div (x^2 - 2x + 1)\).
  • Observe that the correct remainder after the first subtraction is \(x^3 - 5x^2 + 7x - 3\).
  • Compare this to Randy's written step, which shows \(x^3 - 5x^2 + 7x\), indicating he omitted the constant term \(-3\).
  • Identify that this procedural lapse in carrying out the subtraction and tracking terms qualifies as a subtraction error.
  • The final answer is: He makes a subtraction error.

Explanation

  1. Polynomial Long Division Analysis

To identify the error in Randy's division, we first perform the polynomial division of \(2x^4 - 3x^3 - 3x^2 + 7x - 3\) by \(x^2 - 2x + 1\). The correct process is:

  1. Divide the leading term \(2x^4\) by \(x^2\) to get \(2x^2\).
  2. Subtract \(2x^2(x^2 - 2x + 1) = 2x^4 - 4x^3 + 2x^2\) from the original polynomial:

\((2x^4 - 3x^3 - 3x^2 + 7x - 3) - (2x^4 - 4x^3 + 2x^2) = x^3 - 5x^2 + 7x - 3\).

  1. Divide the leading term of the new remainder \(x^3\) by \(x^2\) to get \(x\).
  2. Subtract \(x(x^2 - 2x + 1) = x^3 - 2x^2 + x\) from the previous remainder:

\((x^3 - 5x^2 + 7x - 3) - (x^3 - 2x^2 + x) = -3x^2 + 6x - 3\).

  1. Divide the leading term of the new remainder \(-3x^2\) by \(x^2\) to get \(-3\).
  2. Subtract \(-3(x^2 - 2x + 1) = -3x^2 + 6x - 3\) from the previous remainder to get \(0\).

The correct quotient is \(2x^2 + x - 3\).

  1. Identifying Randy's Error

Now, we compare our correct steps with the steps shown by Randy:

  1. Randy's first subtraction result: \((2x^4 - 3x^3 - 3x^2 + 7x - 3) - (2x^4 - 4x^3 + 2x^2) = x^3 - 5x^2 + 7x\).
  2. Comparing this to the correct remainder \((x^3 - 5x^2 + 7x - 3)\), we see that Randy omitted the constant term \(-3\) when bringing down the terms after the first subtraction.
  3. Because Randy omits this term during the subtraction process, he effectively makes an error in rewriting the problem's intermediate remainder.

Looking at the options, the error isn't explicitly "he forgot to bring down the -3". Let's re-evaluate if any of the given options describe this.
The options are:

  • He makes a subtraction error.
  • He makes an error writing the constant term in the quotient.
  • He makes an error choosing the x-term in the quotient.
  • He makes an error rewriting the problem in long division.

By omitting the -3, the remainder he works with is incorrect. This is a form of an error in the steps of the long division process (specifically in carrying out the subtractions and bringing down terms). Among the options, "He makes a subtraction error" best describes his failure to correctly compute the remainder after subtracting the first term of the product.

  1. Conclusion

The error Randy makes occurs at the first subtraction step. After computing the first part of the quotient (\(2x^2\)), he multiplies and subtracts to find the new remainder. During this step, the term \(-3\) is dropped from the dividend expression in his work, leading to an incomplete representation of the polynomial remainder. This is a procedural error in his subtraction and bookkeeping for the long division. Therefore, he makes a subtraction error.

Examples

Polynomial long division is essential in calculus and engineering for simplifying rational functions, which appear frequently when modeling physical systems like circuits or control theory. For instance, if an engineer needs to analyze a signal represented by a complex rational function, performing long division allows them to decompose the function into a polynomial and a simpler fraction, making it much easier to integrate or perform further analysis. Mastering this technique helps prevent errors in these complex real-world calculations.