Science:Math Exam Resources/Courses/MATH110/December 2013/Question 06 (a)/Solution 1

We first simplify using the exponential rules to get $f(x)=e^{x^{3}+2x^{2}+x-3}.$ Then we can solve for $f(x)=0$.

${\begin{aligned}e^{x^{3}+2x^{2}+x-3}&=0\\x^{3}+2x^{2}+x-3&=\ln(0).\end{aligned}}$

By taking the natural logarithm of both sides, we get a $\ln(0)$ on the right hand side which is bad since $0$ is not in the domain of the natural logarithm. So we conclude that there cannot be a solution to the equation $f(x)=0$. Hence, we have $0$ solutions.