Find the Roots/Zeros Using the Rational Roots Test x^3-5x^2+2x+12

Problem

x3−5*x2+2*x+12=0

Solution

1. Identify the constant term and the leading coefficient. The constant term is (a_0)=12 and the leading coefficient is (a_n)=1

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

p:±1,±2,±3,±4,±6,±12

q:±1

3. Determine the possible rational roots p/q by dividing the factors of p by the factors of q

p/q:±1,±2,±3,±4,±6,±12

4. Test the possible roots using synthetic division or substitution. Testing x=3

(3)3−5*(3)2+2*(3)+12=27−45+6+12=0

Since the result is 0 x=3 is a root and (x−3) is a factor.

5. Divide the polynomial by (x−3) using synthetic division to find the remaining quadratic factor.

(x3−5*x2+2*x+12)÷(x−3)=x2−2*x−4

6. Solve the quadratic equation x2−2*x−4=0 using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(−(−2)±√(,(−2)2−4*(1)*(−4)))/(2*(1))

x=(2±√(,4+16))/2

x=(2±√(,20))/2

x=(2±2√(,5))/2

x=1±√(,5)

Final Answer

x=3,1+√(,5),1−√(,5)

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