- Set up the polynomial division using the coefficients of the dividend \(3x^4 + 20x^3 + 14x^2 + 17x + 30\) and the root of the divisor \(x+6\), which is \(-6\).
- Apply synthetic division to determine the coefficients of the quotient: start with 3, then update each subsequent term by multiplying the previous result by \(-6\) and adding the corresponding dividend coefficient.
- Obtain the resulting sequence \([3, 2, 2, 5]\) with a remainder of 0, which defines the quotient polynomial \(3x^3 + 2x^2 + 2x + 5\).
- Match the result with the given choices, identifying the fourth option as the correct answer. The quotient is \(\boxed{3x^3 + 2x^2 + 2x + 5}\).
Explanation
- Understanding the Problem and Method
We are tasked with dividing the polynomial \(P(x) = 3x^4 + 20x^3 + 14x^2 + 17x + 30\) by the linear divisor \(D(x) = x + 6\). The division of a polynomial of degree 4 by a polynomial of degree 1 will result in a quotient of degree \(4-1=3\). We can use polynomial long division or synthetic division to find the quotient. In this case, since the divisor is of the form \(x - c\) (where \(c = -6\)), synthetic division is a highly efficient and systematic method. We represent the dividend by its coefficients \([3, 20, 14, 17, 30]\) and the divisor by its root \(-6\).
- Performing the Division
To perform the synthetic division with the coefficients \([3, 20, 14, 17, 30]\) and root \(-6\):
- Bring down the first coefficient: \(3\).
- Multiply by the root: \(3 \times (-6) = -18\). Add to the next coefficient: \(20 + (-18) = 2\).
- Multiply by the root: \(2 \times (-6) = -12\). Add to the next coefficient: \(14 + (-12) = 2\).
- Multiply by the root: \(2 \times (-6) = -12\). Add to the next coefficient: \(17 + (-12) = 5\).
- Multiply by the root: \(5 \times (-6) = -30\). Add to the next coefficient: \(30 + (-30) = 0\).
The resulting sequence of values is \([3, 2, 2, 5]\) with a remainder of \(0\). The coefficients \([3, 2, 2, 5]\) correspond to the quotient polynomial \(Q(x) = 3x^3 + 2x^2 + 2x + 5\).
- Finalizing the Result
By comparing our result \(Q(x) = 3x^3 + 2x^2 + 2x + 5\) with the provided options:
- \(3x^2 + 2x + 2\)
- \(3x^3 - 2x^2 + 2x - 5\)
- \(2x^3 + 14x^2 + 17x + 30\)
- \(3x^3 + 2x^2 + 2x + 5\)
We see that the correct option is the fourth one. Therefore, the quotient of the division \((3x^4 + 20x^3 + 14x^2 + 17x + 30) / (x + 6)\) is \(3x^3 + 2x^2 + 2x + 5\).
Examples
Polynomial long division is essential in calculus and engineering for simplifying complex expressions and solving differential equations. For example, when analyzing the motion of an object under varying forces, you might arrive at a complicated rational function representing its velocity; dividing the polynomials allows you to express the velocity as a simpler polynomial part plus a remainder term, which makes integration and limit calculations much more straightforward. Understanding this technique provides the foundational tools for manipulating algebraic structures effectively.