- Divide \(6x^3\) by \(2x\) to get \(3x^2\), then subtract \(3x^2(2x+1)\) from the polynomial to get \(-4x^2 + 8x + 5\).
- Divide \(-4x^2\) by \(2x\) to get \(-2x\), then subtract \(-2x(2x+1)\) to get \(10x + 5\).
- Divide \(10x\) by \(2x\) to get \(5\), then subtract \(5(2x+1)\) to get a remainder of \(0\).
- The quotient is \(\boxed{3x^2 - 2x + 5}\).
Explanation
- Analysis of the Polynomial Division Problem
We are given a cubic polynomial \(P(x) = 6x^3 - x^2 + 8x + 5\) and one of its factors \(Q(x) = 2x + 1\). Our goal is to find the other factor, \(R(x)\), such that \(P(x) = Q(x) \cdot R(x)\). To find this, we perform polynomial long division by dividing the dividend \(6x^3 - x^2 + 8x + 5\) by the divisor \(2x + 1\).
- First Step of Long Division
We start by dividing the leading term of the dividend \(6x^3\) by the leading term of the divisor \(2x\):
\[\frac{6x^3}{2x} = 3x^2\]
This is the first term of our quotient. Now, multiply \(3x^2\) by the divisor \(2x + 1\):
\[3x^2(2x + 1) = 6x^3 + 3x^2\]
Next, subtract this product from the dividend:
\[(6x^3 - x^2 + 8x + 5) - (6x^3 + 3x^2) = -4x^2 + 8x + 5\]
- Second Step of Long Division
Now, take the first term of the new expression \(-4x^2\) and divide it by the leading term of the divisor \(2x\):
\[\frac{-4x^2}{2x} = -2x\]
This is the second term of our quotient. Multiply \(-2x\) by the divisor \(2x + 1\):
\[-2x(2x + 1) = -4x^2 - 2x\]
Subtract this result from the current dividend \(-4x^2 + 8x + 5\):
\[(-4x^2 + 8x + 5) - (-4x^2 - 2x) = 10x + 5\]
- Final Step and Conclusion
Finally, take the first term of the remaining expression \(10x\) and divide it by the leading term of the divisor \(2x\):
\[\frac{10x}{2x} = 5\]
This is the final term of the quotient. Multiply \(5\) by the divisor \(2x + 1\):
\[5(2x + 1) = 10x + 5\]
Subtracting this from the remaining part gives:
\[(10x + 5) - (10x + 5) = 0\]
Since the remainder is 0, the quotient \(R(x)\) is exactly \(3x^2 - 2x + 5\).
Examples
Polynomial long division is a fundamental skill in algebra with practical applications in engineering and computer science. For example, when analyzing the stability of a control system, engineers often need to divide a high-degree polynomial representing the system's characteristic equation by a known factor, such as \((s + p)\), to isolate the roots and determine how the system will react over time. Mastering this process ensures that you can break down complex systems into simpler, manageable parts.