- Divide the leading term \(x^4\) by \(x^2\) to obtain \(x^2\), then subtract to get \(-3x^3 + 7x^2 - 5x + 1\).
- Divide \(-3x^3\) by \(x^2\) to obtain \(-3x\), then subtract to get \(7x^2 - 20x + 1\).
- Divide \(7x^2\) by \(x^2\) to obtain \(7\), then subtract to get the remainder \(-20x + 36\).
- Combine the quotient and the remainder to find the final result \(\boxed{x^2-3x+7-\frac{20x-36}{x^2-5}}\).
Explanation
- Problem Analysis
We are given the rational expression \(\frac{x^4-3x^3+2x^2-5x+1}{x^2-5}\). To perform the division, we divide the polynomial \(P(x) = x^4-3x^3+2x^2-5x+1\) by \(D(x) = x^2-5\) using long division. The goal is to obtain an expression of the form \(Q(x) + \frac{R(x)}{D(x)}\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder.
- First Division Step
First, we divide the leading term of the dividend \(x^4\) by the leading term of the divisor \(x^2\), which gives \(x^2\).
Next, multiply \(x^2\) by the divisor \((x^2-5)\) to get \(x^4-5x^2\).
Subtracting this from the dividend:
\((x^4-3x^3+2x^2-5x+1) - (x^4-5x^2) = -3x^3 + 7x^2 - 5x + 1\).
- Second Division Step
Now, divide the leading term of the new polynomial \(-3x^3\) by \(x^2\) to get \(-3x\).
Multiply \(-3x\) by \((x^2-5)\) to get \(-3x^3+15x\).
Subtracting this from the intermediate polynomial:
\((-3x^3 + 7x^2 - 5x + 1) - (-3x^3 + 15x) = 7x^2 - 20x + 1\).
- Third Division Step and Conclusion
Divide the leading term of the new polynomial \(7x^2\) by \(x^2\) to get 7.
Multiply 7 by \((x^2-5)\) to get \(7x^2-35\).
Subtracting this from the intermediate polynomial:
\((7x^2 - 20x + 1) - (7x^2 - 35) = -20x + 36\).
The quotient is \(Q(x) = x^2-3x+7\) and the remainder is \(R(x) = -20x+36\).
Therefore, the result is \(x^2-3x+7 + \frac{-20x+36}{x^2-5} = x^2-3x+7 - \frac{20x-36}{x^2-5}\).
Examples
Polynomial long division is essential in calculus and algebra for simplifying complex rational functions. For example, when integrating a rational function, breaking it down into a quotient plus a simpler fraction allows the use of standard integration techniques like partial fraction decomposition. This skill helps in fields ranging from engineering to financial modeling, where complex formulas need to be simplified to understand their long-term behavior or to calculate trends accurately.