Use long division to divide the following. Write your remainder as a fraction.
$\left(2 x^3+7 x^2+2 x+9\right) \div(2 x+3)$
We are tasked with dividing the polynomial \(P(x) = 2x^3 + 7x^2 + 2x + 9\) by the linear binomial \(D(x) = 2x + 3\). Our objective is to perform polynomial long division to express the result in the form \(Q(x) + \frac{R}{D(x)}\), where \(Q(x)\) is the quotient and \(R\) is the remainder. Polynomial long division follows a systematic process analogous to numerical long division, where we repeatedly eliminate the highest-degree term of the current dividend.
To begin, we divide the lead term of the dividend \(2x^3\) by the lead term of the divisor \(2x\):
\[\frac{2x^3}{2x} = x^2\]
This gives the first term of our quotient. Next, we multiply this term by the entire divisor:
\[x^2(2x + 3) = 2x^3 + 3x^2\]
Now, we subtract this from our original dividend:
\[(2x^3 + 7x^2 + 2x + 9) - (2x^3 + 3x^2) = 4x^2 + 2x + 9\]
We now have a new intermediate polynomial, \(4x^2 + 2x + 9\).
We repeat the process by dividing the lead term of our current polynomial, \(4x^2\), by the lead term of the divisor \(2x\):
\[\frac{4x^2}{2x} = 2x\]
This is the second term of our quotient. We multiply this by the divisor:
\[2x(2x + 3) = 4x^2 + 6x\]
Subtracting this from our current intermediate polynomial gives:
\[(4x^2 + 2x + 9) - (4x^2 + 6x) = -4x + 9\]
Our new intermediate polynomial is \(-4x + 9\).
Finally, we divide the lead term of the current polynomial, \(-4x\), by the lead term of the divisor \(2x\):
\[\frac{-4x}{2x} = -2\]
This is the final term of our quotient. Multiplying this by the divisor:
\[-2(2x + 3) = -4x - 6\]
Subtracting this from our current intermediate polynomial gives the remainder:
\[(-4x + 9) - (-4x - 6) = 9 + 6 = 15\]
The quotient is \(x^2 + 2x - 2\) and the remainder is \(15\).
Combining our results, we express the division as:
\[\frac{2x^3 + 7x^2 + 2x + 9}{2x + 3} = x^2 + 2x - 2 + \frac{15}{2x + 3}\]
Thus, the result of the division is \(\boxed{x^2 + 2x - 2 + \frac{15}{2x + 3}}\).
Polynomial long division is essential in calculus and algebra for simplifying complex rational functions or finding horizontal/slant asymptotes. For instance, when analyzing the motion of a particle defined by a velocity function \(\frac{2t^3 + 7t^2 + 2t + 9}{2t + 3}\), simplifying this expression helps you identify the long-term trend of the particle's speed by observing the polynomial part \(t^2 + 2t - 2\), while the remainder term represents the influence that decays as time increases. This technique transforms a complicated fraction into a form that is easier to integrate or differentiate.