Solve for x x^3-9x^2+20x-12=0

Problem

x3−9*x2+20*x−12=0

Solution

1. Identify potential rational roots using the Rational Root Theorem. The possible integer roots are factors of the constant term −12 which include ±1,±2,±3,±4,±6,±12

2. Test x=1 by substituting it into the polynomial.

1−9*(1)2+20*(1)−12=1−9+20−12=0

Since the result is 0 (x−1) is a factor.

3. Divide the polynomial x3−9*x2+20*x−12 by (x−1) using synthetic division or long division to find the remaining quadratic factor.

(x3−9*x2+20*x−12)/(x−1)=x2−8*x+12

4. Factor the resulting quadratic expression x2−8*x+12 We look for two numbers that multiply to 12 and add to −8 These numbers are −2 and −6

x2−8*x+12=(x−2)*(x−6)

5. Set each factor equal to zero to find the values of x

(x−1)*(x−2)*(x−6)=0

x−1=0⇒x=1

x−2=0⇒x=2

x−6=0⇒x=6

Final Answer

x=1,2,6

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