- Identify the coefficients \(2, 4, 2, 8, 8\) and the divisor value \(c = -2\).
- Apply the synthetic division process to generate the quotient coefficients \(2, 0, 2, 4\) and a remainder of \(0\).
- Construct the final quotient polynomial \(2x^3 + 2x + 4\).
- Select the first option as the correct result: \(\boxed{2x^3 + 2x + 4}\).
Explanation
- Analysis of the Polynomial Division Problem
We are tasked with dividing the polynomial \(P(x) = 2x^4 + 4x^3 + 2x^2 + 8x + 8\) by the linear binomial \(x + 2\) using synthetic division. Synthetic division is a streamlined technique for dividing a polynomial by a divisor of the form \(x - c\). In this case, since the divisor is \(x + 2\), we identify \(c = -2\). The coefficients of our dividend polynomial \(P(x)\) are \(2, 4, 2, 8, 8\).
- Performing Synthetic Division
To perform synthetic division, we set up a tableau with the value \(c = -2\) on the left and the coefficients of \(P(x)\) on the right:
\[ \begin{array}{c|rrrrr} -2 & 2 & 4 & 2 & 8 & 8 \ & & & & & \\ & & & & & \end{array} \]
- Bring down the first coefficient, which is \(2\).
- Multiply this by \(c = -2\): \(2 \times (-2) = -4\). Write this under the second coefficient and add: \(4 + (-4) = 0\).
- Multiply the new value \(0\) by \(c = -2\): \(0 \times (-2) = 0\). Write this under the third coefficient and add: \(2 + 0 = 2\).
- Multiply this value \(2\) by \(c = -2\): \(2 \times (-2) = -4\). Write this under the fourth coefficient and add: \(8 + (-4) = 4\).
- Multiply the value \(4\) by \(c = -2\): \(4 \times (-2) = -8\). Write this under the final coefficient and add: \(8 + (-8) = 0\).
The resulting values at the bottom of the tableau are \(2, 0, 2, 4\) with a remainder of \(0\).
- Conclusion and Identification of the Correct Option
The resulting coefficients \(2, 0, 2, 4\) represent the quotient polynomial, where the degree is one less than the original polynomial. Since the original polynomial was of degree 4, the quotient is of degree 3:
\[Q(x) = 2x^3 + 0x^2 + 2x + 4 = 2x^3 + 2x + 4.\]
Since the remainder is \(0\), the division is exact. Comparing this result with the given choices:
- \(2x^3 + 2x + 4\)
- \(2x^4 + 2x^2 + 4x\)
- \(2x^3 + 2x + 4 + \frac{1}{x+2}\)
- \(2x^3 + 2x^2 + 4\)
The correct quotient is \(2x^3 + 2x + 4\).
Examples
Synthetic division is a highly efficient way to simplify polynomials, which is a common task in fields such as engineering, computer science, and physics. For example, if you are modeling the trajectory of an object where forces change based on specific time intervals, you might encounter polynomial equations representing position or velocity. Using synthetic division allows you to quickly factor these equations to find specific time points where the object passes through a certain position, helping in the design of efficient control systems or game physics engines.