Answered

What is the quotient of $\left(2 x^4-3 x^3-3 x^2+7 x-3\right) \div\left(x^2-2 x+1\right)$?


A$2 x^2+x-3$
B$2 x^2-x$
C$2 x^2+x-3-\frac{6}{x^2-2 x+1}$
D$2 x^4-3 x^3-\frac{11}{2}$

Verified Answer :

  • Simplify the dividend and divisor polynomials for division: \(P(x) = 2x^4 - 3x^3 - 3x^2 + 7x - 3\) and \(D(x) = x^2 - 2x + 1\).
  • Perform polynomial long division to divide \(P(x)\) by \(D(x)\).
  • Recognize that the quotient is exactly \(2x^2 + x - 3\) with no remainder.
  • Select the matching option: \(\boxed{a}\).

Explanation

  1. Problem Analysis

The objective is to find the quotient of the division \((2x^4 - 3x^3 - 3x^2 + 7x - 3) \div (x^2 - 2x + 1)\). We begin by identifying the dividend, \(P(x) = 2x^4 - 3x^3 - 3x^2 + 7x - 3\), and the divisor, \(D(x) = x^2 - 2x + 1\). Note that the divisor can be factored as \(D(x) = (x-1)^2\).

  1. Performing Polynomial Division

To find the quotient, we perform polynomial long division of \(P(x)\) by \(D(x)\).

  1. Divide the leading term of the dividend \(2x^4\) by the leading term of the divisor \(x^2\), giving \(2x^2\).
  2. Multiply \(2x^2\) by \((x^2 - 2x + 1)\) to get \(2x^4 - 4x^3 + 2x^2\).
  3. Subtract this from the dividend: \((2x^4 - 3x^3 - 3x^2 + 7x - 3) - (2x^4 - 4x^3 + 2x^2) = x^3 - 5x^2 + 7x - 3\).
  4. Divide the new leading term \(x^3\) by \(x^2\), giving \(+x\).
  5. Multiply \(x\) by \((x^2 - 2x + 1)\) to get \(x^3 - 2x^2 + x\).
  6. Subtract this: \((x^3 - 5x^2 + 7x - 3) - (x^3 - 2x^2 + x) = -3x^2 + 6x - 3\).
  7. Divide \(-3x^2\) by \(x^2\), giving \(-3\).
  8. Multiply \(-3\) by \((x^2 - 2x + 1)\) to get \(-3x^2 + 6x - 3\).
  9. Subtracting this leaves a remainder of \(0\).
  1. Conclusion

The result of the division is the quotient \(2x^2 + x - 3\), with a remainder of \(0\). Comparing this to the given choices, choice (a) matches our result exactly. While the original problem expression provided contained a potential transcription error (simplifying to \(-4x^2 + 7x - 3\)), the intended division problem leading to the provided options is clearly \((2x^4 - 3x^3 - 3x^2 + 7x - 3) \div (x^2 - 2x + 1)\).

Examples

Polynomial division is essential in algebra and calculus for simplifying rational functions, which are used to model real-world scenarios such as drug concentration levels over time in the bloodstream or the trajectory of an object under varying resistance. By dividing complex polynomials, we can better understand the behavior and limits of these functions, allowing for accurate predictions and optimizations in fields ranging from engineering to economics.

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