- Identify the dividend \(8x^3 - x^2 + 6x + 7\) and the divisor \(2x - 1\).
- Recognize that the leading term of the dividend is \(8x^3\) and the leading term of the divisor is \(2x\).
- Perform the first step of polynomial long division by dividing the leading term of the dividend by the leading term of the divisor: \(\frac{8x^3}{2x} = 4x^2\).
- The correct first step is to divide \(8x^3\) by \(2x\).
- The final answer is the first option: \(\boxed{\text{Divide } 8x^3 \text{ by } 2x}\).
Explanation
- Understanding the Problem
To solve the polynomial division problem \((8x^3 - x^2 + 6x + 7) \div (2x - 1)\), we follow the standard algorithm for polynomial long division. The goal is to determine the first term of the quotient, which is found by dividing the highest-degree term of the dividend by the highest-degree term of the divisor. The dividend is the polynomial being divided, \(8x^3 - x^2 + 6x + 7\), and the divisor is the polynomial we are dividing by, \(2x - 1\).
- Identifying Leading Terms
The standard process for polynomial long division starts by focusing on the leading terms of both the dividend and the divisor. The leading term is the term with the highest power of the variable \(x\). In our case:
- The leading term of the dividend \(8x^3 - x^2 + 6x + 7\) is \(8x^3\).
- The leading term of the divisor \(2x - 1\) is \(2x\).
- Performing the First Step
The first step in the long division algorithm is to divide the leading term of the dividend by the leading term of the divisor to obtain the first term of the quotient.
Performing this division:
\[\frac{8x^3}{2x} = 4x^2\]
This identifies that the first term of our quotient is \(4x^2\). Comparing this to the given options, the initial action performed is dividing \(8x^3\) by \(2x\).
- Conclusion
We have determined that the correct procedure is to divide the highest-degree term of the dividend, \(8x^3\), by the highest-degree term of the divisor, \(2x\). Therefore, the correct choice among the provided options is the first one. The final answer is the first option.
Examples
Polynomial long division is essential in fields like engineering and physics for simplifying complex rational expressions. For instance, when designing a control system, engineers often need to divide a system's output polynomial by its input polynomial to find the transfer function. By mastering the division of leading terms, you are learning the fundamental technique for breaking down and analyzing the behavior of complex dynamic systems.