- Perform polynomial long division by dividing the leading terms to find the quotient.
- Subtract the product of the divisor and quotient terms from the dividend to find the remainder.
- The quotient is \(\(2x + 1\)\) and the remainder is \(\(2x - 5\)\).
- The final expression is \(\(2x + 1 + \frac{2x - 5}{x^2 + x + 1}\)\), which matches option c.
- Final Answer: \(\boxed{c}\)
Explanation
- Problem Analysis
The objective is to divide the polynomial \(2x^3 + 3x^2 + 5x - 4\) by \(x^2 + x + 1\) and express the result in the form \(Q(x) + \frac{R(x)}{D(x)}\), where \(Q(x)\) is the quotient, \(R(x)\) is the remainder, and \(D(x)\) is the divisor. This process is known as polynomial long division.
- Polynomial Long Division Process
To perform the division \((2x^3 + 3x^2 + 5x - 4) \div (x^2 + x + 1)\):
- Divide the leading term of the dividend (\(2x^3\)) by the leading term of the divisor (\(x^2\)) to get the first term of the quotient: \(2x^3 / x^2 = 2x\).
- Multiply the entire divisor by this term: \(2x(x^2 + x + 1) = 2x^3 + 2x^2 + 2x\).
- Subtract this product from the original dividend: \((2x^3 + 3x^2 + 5x - 4) - (2x^3 + 2x^2 + 2x) = x^2 + 3x - 4\).
- Divide the leading term of this new expression (\(x^2\)) by the leading term of the divisor (\(x^2\)) to get the next term of the quotient: \(x^2 / x^2 = 1\).
- Multiply the divisor by this new term: \(1(x^2 + x + 1) = x^2 + x + 1\).
- Subtract this product from the current expression: \((x^2 + 3x - 4) - (x^2 + x + 1) = 2x - 5\).
- Since the degree of the remainder (\(2x - 5\)) is less than the degree of the divisor (\(x^2 + x + 1\)), the division process terminates. The quotient is \(2x + 1\) and the remainder is \(2x - 5\).
- Conclusion and Comparison
Combining the results, the expression can be written as:
\(\frac{2x^3 + 3x^2 + 5x - 4}{x^2 + x + 1} = (2x + 1) + \frac{2x - 5}{x^2 + x + 1}\).
Comparing this with the given options:
A\(2x + 1 + \frac{4x - 4}{x^2 + x + 1}\)
B\(2x + 5 + \frac{12x + 1}{x^2 + x + 1}\)
C\(2x + 1 + \frac{2x - 5}{x^2 + x + 1}\)
D\(2x + 1 + \frac{2x - 1}{x^2 + x + 1}\)
The result matches option c.
Examples
Polynomial division is a fundamental skill in algebra with applications in fields like engineering and physics. For instance, when analyzing the motion of an object under a complex force field, the position or velocity might be represented by a rational function. Simplifying such expressions through polynomial division allows for easier integration or differentiation, helping scientists predict an object's trajectory more accurately. Understanding how to break down complex ratios into a simpler polynomial component and a manageable fraction is an essential tool for any quantitative analysis.