15. the function f is given by f(x) = (6x^2+ax + 2)/(x + 3) and has a slant asymptote of y = 6x+3. what is…

Explanation:

Step1: Divide polynomials

When we divide the numerator $6x^{2}+ax + 2$ by the denominator $x + 3$ using polynomial long - division or synthetic division. The quotient of the division of a quadratic polynomial $Ax^{2}+Bx + C$ by a linear polynomial $x + k$ gives a linear polynomial of the form $mx + n$. For the rational function $f(x)=\frac{6x^{2}+ax + 2}{x + 3}$, using polynomial long - division: $6x^{2}+ax + 2=(x + 3)(6x + b)+r$. Expanding the right - hand side gives $6x^{2}+ax + 2=6x^{2}+(b + 18)x+(3b + r)$. We know that the slant asymptote is found by performing the division of the numerator by the denominator and ignoring the remainder. When we divide $6x^{2}+ax + 2$ by $x + 3$, the quotient is the equation of the slant asymptote. Dividing $6x^{2}+ax + 2$ by $x + 3$ using polynomial long - division: [ \begin{align*} \frac{6x^{2}+ax + 2}{x+3}&=6x+(a - 18)+\frac{56 - 3a}{x + 3} \end{align*} ] The equation of the slant asymptote is $y = 6x+(a - 18)$.

Step2: Equate to given slant asymptote

Since the slant asymptote is $y=6x + 3$, we set the non - remainder part of the quotient equal to the given slant asymptote. We have $a-18 = 3$.

Step3: Solve for $a$

Adding 18 to both sides of the equation $a-18 = 3$, we get $a=21$.

Answer:

D. 21