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±\frac{5}{2},±5,±\frac{1}{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 -5 and q divides the leading coefficient 2. List all candidates \frac{p}{q}.
z=\frac{1}{2}
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.
z^{2}+2z+5=0
By Factor theorem, z-k is a factor of the polynomial for each root k. Divide 2z^{3}+3z^{2}+8z-5 by 2\left(z-\frac{1}{2}\right)=2z-1 to get z^{2}+2z+5. Solve the equation where the result equals to 0.
z=\frac{-2±\sqrt{2^{2}-4\times 1\times 5}}{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 5 for c in the quadratic formula.
z=\frac{-2±\sqrt{-16}}{2}
Do the calculations.
z\in \emptyset
Since the square root of a negative number is not defined in the real field, there are no solutions.
z=\frac{1}{2}
List all found solutions.