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±72,±36,±24,±18,±12,±9,±8,±6,±4,±3,±2,±1
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -72 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.
a=-3
Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.
a^{2}+2a-24=0
By Factor theorem, a-k is a factor of the polynomial for each root k. Divide a^{3}+5a^{2}-18a-72 by a+3 to get a^{2}+2a-24. Solve the equation where the result equals to 0.
a=\frac{-2±\sqrt{2^{2}-4\times 1\left(-24\right)}}{2}
All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 2 for b, and -24 for c in the quadratic formula.
a=\frac{-2±10}{2}
Do the calculations.
a=-6 a=4
Solve the equation a^{2}+2a-24=0 when ± is plus and when ± is minus.
a=-3 a=-6 a=4
List all found solutions.