- The polynomial \(P(x) = x^3 + 10x^2 + 21x + 16\) is divided by \(x+2\).
- Using synthetic division with root \(c = -2\), the coefficients are processed: \(1, 10, 21, 16\).
- The resulting coefficients of the quotient are \(1, 8, 5\) and the remainder is \(6\).
- The quotient is \(x^2 + 8x + 5\) and the remainder is \(6\).
- Final Answer: Quotient is \(\boxed{x^2+8x+5}\), Remainder is \(\boxed{6}\).
Explanation
- Analyzing the Polynomial Division Problem
We are tasked with dividing the polynomial \(P(x) = x^3 + 10x^2 + 21x + 16\) by the linear factor \(x + 2\) using synthetic division. Synthetic division is an efficient shorthand for long division when the divisor is of the form \(x - c\). In our case, the divisor is \(x + 2\), which corresponds to \(x - (-2)\), so the value we use for our synthetic division is \(c = -2\). The coefficients of our polynomial \(P(x)\) are \(1, 10, 21,\) and \(16\).
- Performing Synthetic Division
To perform synthetic division, we set up a table using the coefficients of \(P(x)\) and the root \(c = -2\).
- We write the coefficients: \(1, 10, 21, 16\).
- We bring down the first coefficient, which is \(1\).
- We multiply this by the root \(-2\), resulting in \(-2\), and add this to the next coefficient \(10\), giving \(10 - 2 = 8\).
- We multiply the new value \(8\) by the root \(-2\), giving \(-16\), and add this to the next coefficient \(21\), resulting in \(21 - 16 = 5\).
- Finally, we multiply the new value \(5\) by the root \(-2\), giving \(-10\), and add this to the constant term \(16\), resulting in \(16 - 10 = 6\).
- Identifying the Quotient and Remainder
From the synthetic division table, the final row of values is \(1, 8, 5, 6\). The first three values, \(1, 8,\) and \(5\), represent the coefficients of the quotient polynomial. Since the original dividend was a third-degree polynomial and we divided by a first-degree factor, the quotient will be a second-degree polynomial. Therefore, the quotient is \(1x^2 + 8x + 5\). The last value, \(6\), is the remainder of the division.
Examples
Synthetic division is used in computer graphics and game development when dealing with complex geometric shapes defined by polynomials. For instance, if an object's trajectory is modeled by a high-degree polynomial \(P(x)\) and you need to determine the intersection points with a plane (a simple linear equation \(x-c\)), you would perform synthetic division to find the reduced equation. This simplification makes it much faster for computers to calculate intersections and render the scene smoothly in real-time.