- Identify the dividend coefficients as \(5, 19, 6, -18\) and the divisor root as \(-3\).
- Execute synthetic division by bringing down the leading coefficient, multiplying by the root, adding to the next coefficient, and repeating the process.
- Determine the quotient coefficients as \(5, 4, -6\), which yields the polynomial \(W(x) = 5x^2 + 4x - 6\).
- Select the matching option (c), concluding the width is \(\boxed{5x^2 + 4x - 6}\).
Explanation
- Analysis of the Problem
To find the width \(W(x)\) of the rectangle, we know that the area \(A(x)\) is the product of the length \(L(x)\) and the width \(W(x)\). Thus, we have \(A(x) = L(x) \cdot W(x)\), which means the width is given by the quotient of the division \(W(x) = \frac{A(x)}{L(x)} = \frac{5x^3 + 19x^2 + 6x - 18}{x + 3}\). We are tasked with performing this division using the synthetic division method.
- Performing Synthetic Division
The dividend is the polynomial \(A(x) = 5x^3 + 19x^2 + 6x - 18\), and its coefficients are \(5, 19, 6, -18\). The divisor is \(L(x) = x + 3\), so the root to be used in the synthetic division is \(x = -3\). We set up the synthetic division as follows:
- Write down the coefficients \(5, 19, 6, -18\) in a row.
- Place the root \(-3\) to the left.
- Bring down the first coefficient, which is 5.
- Multiply this 5 by \(-3\) to get -15, and add it to the next coefficient, 19. The result is \(19 - 15 = 4\).
- Multiply this 4 by \(-3\) to get -12, and add it to the next coefficient, 6. The result is \(6 - 12 = -6\).
- Multiply this -6 by \(-3\) to get 18, and add it to the last constant coefficient, -18. The result is \(-18 + 18 = 0\).
- Conclusion
The numbers obtained at the bottom of the division are \(5, 4, -6\), with a remainder of 0. These coefficients correspond to the quotient polynomial, where the degrees are one less than the degrees of the original polynomial. Therefore, the resulting width is \(W(x) = 5x^2 + 4x - 6\). This matches option (c).
Examples
Synthetic division is a shorthand method of polynomial division, particularly useful when dividing by a linear factor of the form \(x - c\). In real-world applications, this method is used in fields such as engineering, economics, and physics for simplifying complex models. For example, if a company models its total revenue as a cubic function \(R(x)\) and needs to find the average revenue per unit \(A(x) = \frac{R(x)}{x+k}\), synthetic division provides a rapid way to calculate this quotient, helping in efficient resource management and business forecasting.