Solve p^3-3p-52=0

$$p ^ { 3 } - 3 p - 52 = 0$$

Solve for p

$$p=4$$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -52 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±52,±26,±13,±4,±2,±1$$

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.

$$p=4$$

By Factor theorem, p-k is a factor of the polynomial for each root k. Divide p^{3}-3p-52 by p-4 to get p^{2}+4p+13. Solve the equation where the result equals to 0.

$$p^{2}+4p+13=0$$

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, 4 for b, and 13 for c in the quadratic formula.

$$p=\frac{-4±\sqrt{4^{2}-4\times 1\times 13}}{2}$$

Do the calculations.

$$p=\frac{-4±\sqrt{-36}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$p\in \emptyset$$

List all found solutions.

$$p=4$$