- Start long division by dividing the leading term of the dividend by the leading term of the divisor to get \(x^3\).
- Subtract the product \(x^3(x^2 - 5)\) from the dividend to obtain the remainder \(2x^3 - 3x^2 - 10x + 15\).
- Repeat the process by dividing the leading terms to find the subsequent quotient terms \(2x\) and \(-3\).
- Conclude that the resulting quotient is \(\boxed{x^3 + 2x - 3}\).
Explanation
- Performing the First Step of Long Division
To find the quotient of the polynomial \(P(x) = x^5 - 3x^3 - 3x^2 - 10x + 15\) and the divisor \(D(x) = x^2 - 5\), we perform polynomial long division.
First, we divide the highest degree term of \(P(x)\), which is \(x^5\), by the highest degree term of \(D(x)\), which is \(x^2\).
\[\frac{x^5}{x^2} = x^3\]
This gives us the first term of our quotient. Next, we multiply \(x^3\) by the divisor \((x^2 - 5)\):
\[x^3(x^2 - 5) = x^5 - 5x^3\]
Now, we subtract this from the original polynomial \(P(x)\):
\[(x^5 - 3x^3 - 3x^2 - 10x + 15) - (x^5 - 5x^3) = 2x^3 - 3x^2 - 10x + 15\]
This is our new remainder.
- Performing the Second Step of Long Division
We continue the process by dividing the leading term of the new remainder, \(2x^3\), by the leading term of the divisor, \(x^2\):
\[\frac{2x^3}{x^2} = 2x\]
This is the second term of our quotient. We multiply \(2x\) by the divisor \((x^2 - 5)\):
\[2x(x^2 - 5) = 2x^3 - 10x\]
Subtracting this from our current remainder:
\[(2x^3 - 3x^2 - 10x + 15) - (2x^3 - 10x) = -3x^2 + 15\]
This is our next remainder.
- Finalizing the Long Division and Conclusion
Finally, we divide the leading term of the remainder \(-3x^2\) by the leading term of the divisor \(x^2\):
\[\frac{-3x^2}{x^2} = -3\]
This is the final term of our quotient. We multiply \(-3\) by the divisor \((x^2 - 5)\):
\[-3(x^2 - 5) = -3x^2 + 15\]
Subtracting this from our current remainder:
\[(-3x^2 + 15) - (-3x^2 + 15) = 0\]
Since the remainder is \(0\), the division is exact. Combining the terms \(x^3\), \(2x\), and \(-3\), the quotient is \(x^3 + 2x - 3\).
Examples
Polynomial division is widely used in engineering and physics, for example, when analyzing the motion of an object under varying forces or in signal processing to simplify complex transfer functions. If an engineer has a complex system response modeled by a high-degree polynomial, they can divide it by a simpler characteristic polynomial to isolate the specific behavior of the system, making it easier to analyze and control. This demonstrates how abstract algebraic techniques are applied to solve real-world technical problems.