- Identify the coefficients of the dividend as \(3, 6, 2, 9, 10\) and the root as \(-2\) from the divisor \(x+2\).
- Apply synthetic division to obtain the coefficients of the quotient: \(3, 0, 2, 5\) with a remainder of \(0\).
- The resulting polynomial is \(3x^3 + 2x + 5\).
- Select the corresponding option: \(\boxed{b}\).
Explanation
- Analysis of the Polynomial Division Problem
To solve the polynomial division \((3x^4 + 6x^3 + 2x^2 + 9x + 10) \div (x + 2)\), we use synthetic division. The divisor is \(x + 2\), which corresponds to the root \(x = -2\). The coefficients of the dividend polynomial are \(3, 6, 2, 9, 10\).
- Performing Synthetic Division
We set up the synthetic division using the root \(x = -2\) and the coefficients \(3, 6, 2, 9, 10\):
- Start by bringing down the first coefficient: 3.
- Multiply the root -2 by 3 to get -6, then add to the next coefficient: \(6 + (-6) = 0\).
- Multiply -2 by 0 to get 0, then add to the next coefficient: \(2 + 0 = 2\).
- Multiply -2 by 2 to get -4, then add to the next coefficient: \(9 + (-4) = 5\).
- Multiply -2 by 5 to get -10, then add to the next coefficient: \(10 + (-10) = 0\).
The resulting values are \(3, 0, 2, 5\) with a remainder of 0.
- Interpreting the Result
The coefficients of the quotient are \(3, 0, 2, 5\), which represent the polynomial \(3x^3 + 0x^2 + 2x + 5\), or simply \(3x^3 + 2x + 5\). Since the remainder is 0, the division is exact. Looking at the given options:
A\(3x^3 + 12x^2 + 26x + 61 + \frac{132}{x-2}\)
B\(3x^3 + 2x + 5\)
C\(3x^3 + 12x^2 + 26x + 61 + \frac{132}{x+2}\)
D\(3x^4 + 2x^2 + 5x\)
The correct quotient is option b.
Examples
Polynomial division is commonly used in engineering and computer graphics to simplify complex equations. For example, when analyzing the motion of a robot arm or rendering 3D objects, you might need to find the roots of high-degree polynomials by dividing them by known factors to simplify the calculation, allowing for more efficient real-time processing and control.