- Identify the dividend from the top row coefficients: \(2x^3 + 11x^2 + 18x + 9\).
- Determine the divisor from the root value \(-3\): \(x + 3\).
- Extract the quotient coefficients from the bottom row: \(2x^2 + 5x + 3\).
- The identified components are: Divisor \(\boxed{x + 3}\), Dividend \(\boxed{2x^3 + 11x^2 + 18x + 9}\), Quotient \(\boxed{2x^2 + 5x + 3}\).
Explanation
- Identifying the Dividend
Synthetic division is a shorthand method of polynomial division. The top row of the table represents the coefficients of the dividend polynomial. Since the coefficients are 2, 11, 18, and 9, and there are 4 terms, the dividend is a cubic polynomial:
\[2x^3 + 11x^2 + 18x + 9\]
- Identifying the Divisor
The number to the left of the vertical bar represents the root of the divisor. In this case, the root is \(-3\). If a polynomial is divided by a linear factor of the form \((x - c)\), the root used in the synthetic division is \(c\). Here, \(c = -3\), so the divisor is:
\[x - (-3) = x + 3\]
- Identifying the Quotient
The bottom row, excluding the remainder, provides the coefficients of the quotient. With coefficients 2, 5, and 3, and since the original dividend was a cubic polynomial divided by a linear polynomial, the quotient must be a quadratic polynomial:
\[2x^2 + 5x + 3\]
The final term in the bottom row is 0, which confirms there is no remainder for this division.
- Conclusion
Based on the analysis of the synthetic division table:
- The Divisor is \(x + 3\) (option d).
- The Dividend is \(2x^3 + 11x^2 + 18x + 9\).
- The Quotient is \(2x^2 + 5x + 3\).
Thus, the correct components are:
Divisor: \(\boxed{x + 3}\), Dividend: \(\boxed{2x^3 + 11x^2 + 18x + 9}\), Quotient: \(\boxed{2x^2 + 5x + 3}\).
Examples
Synthetic division is a highly efficient tool for engineers and computer scientists when working with polynomials, such as in signal processing or control theory, where you might need to find the roots of a high-degree polynomial. By simplifying the division process, one can quickly factorize equations or evaluate functions at specific points, making complex calculations more manageable and reducing the likelihood of manual errors. Understanding this shortcut allows for faster analysis of systems described by polynomial models.