- Identify the coefficients of the dividend \(P(x) = 2x^3 + 4x^2 - 35x + 15\) as 2, 4, -35, and 15 and the divisor root \(c = 3\).
- Apply the synthetic division method by bringing down the leading coefficient, multiplying by \(c\), and adding systematically.
- Interpret the result: the coefficients 2, 10, -5 yield the quotient \(2x^2 + 10x - 5\), with a remainder of 0.
- The final answer is \(\boxed{c}\).
Explanation
- Analysis of the Polynomial Division Problem
The problem requires us to divide the cubic polynomial \(P(x) = 2x^3 + 4x^2 - 35x + 15\) by the linear binomial \(x - 3\) using synthetic division. Synthetic division is an efficient method for dividing a polynomial by a linear divisor of the form \(x - c\). In this case, \(c = 3\). The coefficients of the polynomial are 2, 4, -35, and 15.
- Performing the Synthetic Division
To perform the synthetic division, we set up the coefficients and the divisor constant \(c=3\):
- Place \(c=3\) to the left.
- List the coefficients \(2, 4, -35, 15\) to the right.
The process is as follows:
- Bring down the first coefficient, 2.
- Multiply 2 by \(c=3\) to get 6. Add this to the next coefficient (4): \(4 + 6 = 10\).
- Multiply the result 10 by \(c=3\) to get 30. Add this to the next coefficient (-35): \(-35 + 30 = -5\).
- Multiply the result -5 by \(c=3\) to get -15. Add this to the final coefficient (15): \(15 + (-15) = 0\).
The numbers at the bottom are 2, 10, -5, and 0. The last number, 0, represents the remainder, while the preceding numbers (2, 10, -5) are the coefficients of the resulting quotient polynomial.
- Identifying the Quotient and Final Result
Since we started with a cubic polynomial (\(x^3\)) and divided by a linear polynomial (\(x^1\)), the quotient must be a quadratic polynomial (\(x^2\)). Using the coefficients obtained, the quotient is \(2x^2 + 10x - 5\). Comparing this result with the given options, it corresponds to option c.
Examples
Synthetic division is widely used in computer science and engineering, particularly when evaluating polynomials at specific points or when building algorithms for numerical analysis. For instance, in graphics rendering, calculating values of curves represented by polynomials requires efficient division and evaluation methods. This method simplifies complex calculations, making it an essential tool for programmers designing high-performance software.