Find the Roots/Zeros Using the Rational Roots Test x^3-4x^2-25x+100

Problem

x3−4*x2−25*x+100=0

Solution

1. Identify the constant term (a_0)=100 and the leading coefficient (a_n)=1

2. List all possible factors of the constant term p and the leading coefficient q

p∈{±1,±2,±4,±5,±10,±20,±25,±50,±100}

q∈{±1}

3. Determine the set of possible rational roots p/q

p/q∈{±1,±2,±4,±5,±10,±20,±25,±50,±100}

4. Test values using synthetic division or substitution. Testing x=4

4−4*(4)−25*(4)+100=64−64−100+100=0

5. Divide the polynomial by the factor (x−4) to find the remaining quadratic.

(x3−4*x2−25*x+100)÷(x−4)=x2−25

6. Solve the resulting quadratic equation x2−25=0

x2=25

x=±5

Final Answer

x3−4*x2−25*x+100=0⇒x∈{4,5,−5}

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