<s>[INST] Finding the zeroes in a function I have come across a problem on my trigonometry homework where we need to find the zeros of a function without the use of a calculator. 
The Equation:
Given one of the zeros is $x=5$
$f(x) = x^3+2x^2-23x-60$
Is there a more efficient way to go about this problem than simply factor it out?
 [/INST] Yes if one of the roots is $5$ it means $x-5$ is factor of the given polynomial.Dividing the polynomial by $x-5$ we get the quadratic equation $x^2+7x+12$. Now the roots of this quadratic equation are $\frac{-7+(49-4*12)^{\frac{1}{2}}}{2}$ and $\frac{-7-(49-4*12)^{\frac{1}{2}}}{2}$. Thus the remaining two roots are $-4$ and $-3$. In general if we have a quadratic equation $ax^2+bx+c$ it has roots $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.
For dividing $x^3+2x^2-23x-60$ by $x-5$ take a look at Polynomial Long Division.
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