Find the Roots/Zeros Using the Rational Roots Test x^4-5x^2-36

Problem

x^4−5*x^2−36=0

Solution

1. Identify the constant term (a_0)=−36 and the leading coefficient (a_n)=1

2. List all possible factors of the constant term p ±1,±2,±3,±4,±6,±9,±12,±18,±36

3. List all possible factors of the leading coefficient q ±1

4. Determine the possible rational roots p/q ±1,±2,±3,±4,±6,±9,±12,±18,±36

5. Test the possible roots using synthetic division or substitution.

6. Substitute x=3 3^4−5*(3^2)−36=81−45−36=0 Thus, x=3 is a root.

7. Substitute x=−3 (−3)^4−5*(−3)^2−36=81−45−36=0 Thus, x=−3 is a root.

8. Factor the polynomial using the known roots: (x−3)*(x+3)*(x^2+4)=0

9. Solve the remaining quadratic factor x^2+4=0 to find the non-rational roots.

10. Calculate x^2=−4 which gives x=±2*i

11. Identify the rational roots from the complete set of zeros.

Final Answer

x=±3

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