Verified Answer :
- Identify the root of the divisor as \(y = -2\).
- Apply synthetic division with the coefficients \(1, 11, -18, 6\) to find the quotient coefficients \(1, 9, -36\) and remainder \(78\).
- Write the final result as the quotient plus the remainder over the divisor: \(\boxed{y^2 + 9y - 36 + \frac{78}{y + 2}}\).
Explanation
- Problem Analysis and Method Selection
The goal is to divide the cubic polynomial \(P(y) = y^3 + 11y^2 - 18y + 6\) by the linear binomial \(D(y) = y + 2\). We can solve this using polynomial long division or synthetic division. Synthetic division is particularly efficient when dividing by a linear factor of the form \(y - k\). Here, our divisor is \(y + 2\), which is equivalent to \(y - (-2)\), so we will use the constant \(k = -2\).
- Performing the Division
We set up synthetic division with the root \(k = -2\) and the coefficients of the polynomial \(y^3 + 11y^2 - 18y + 6\), which are \(1, 11, -18,\) and \(6\).
- Bring down the first coefficient: \(1\).
- Multiply this by the root \(-2\) to get \(-2\), and add it to the second coefficient \(11\): \(11 + (-2) = 9\).
- Multiply this result \(9\) by the root \(-2\) to get \(-18\), and add it to the third coefficient \(-18\): \(-18 + (-18) = -36\).
- Multiply this result \(-36\) by the root \(-2\) to get \(72\), and add it to the constant term \(6\): \(6 + 72 = 78\).
The numbers below the line are \(1, 9, -36,\) and the remainder \(78\).
- Conclusion and Verification
The coefficients \(1, 9, -36\) represent the quotient polynomial of degree 2, which is \(Q(y) = y^2 + 9y - 36\). The final value \(78\) is the remainder. Therefore, the result of the division can be expressed as:
\[ \frac{y^3 + 11y^2 - 18y + 6}{y + 2} = y^2 + 9y - 36 + \frac{78}{y + 2} \]
We can verify this by multiplying the quotient by the divisor and adding the remainder:
\[ (y^2 + 9y - 36)(y + 2) + 78 = (y^3 + 2y^2 + 9y^2 + 18y - 36y - 72) + 78 = y^3 + 11y^2 - 18y + 6 \]
The result is correct. The final answer for the division is \(\boxed{y^2 + 9y - 36 + \frac{78}{y + 2}}\).
Examples
Polynomial division is a fundamental skill in algebra, useful for tasks like simplifying rational expressions, finding the roots of polynomials, or performing partial fraction decomposition in calculus. For example, if an engineer is modeling the efficiency of a machine over time with a cubic function \(P(t)\) and needs to normalize it by a factor \(t+c\), performing this division allows them to extract a simpler quadratic model plus a remainder term, which might represent a steady-state offset or a transient behavior that fades out over time. Understanding how to break down complex expressions into simpler components is essential in fields ranging from structural engineering to data analysis.