Find the Roots/Zeros Using the Rational Roots Test x^3+7x^2-16x+18

Problem

x3+7*x2−16*x+18

Solution

1. Identify the constant term (a_0)=18 and the leading coefficient (a_n)=1

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

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

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

5. Test the values using synthetic division or substitution into the polynomial ƒ(x)=x3+7*x2−16*x+18

6. Evaluate ƒ(1)=1+7−16+18=10

7. Evaluate ƒ*(−1)=−1+7+16+18=40

8. Evaluate ƒ(2)=8+28−32+18=22

9. Evaluate ƒ*(−2)=−8+28+32+18=70

10. Evaluate ƒ(3)=27+63−48+18=60

11. Evaluate ƒ*(−3)=−27+63+48+18=102

12. Evaluate ƒ*(−9)=−729+567+144+18=0

13. Identify x=−9 as a rational root.

14. Divide the polynomial by (x+9) to find the remaining quadratic factor: x2−2*x+2

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

16. Calculate x=(2±√(,4−8))/2=(2±√(,−4))/2=1±i

17. Conclude that the only rational root is −9 while the other roots are complex.

Final Answer

Roots of *x3+7*x2−16*x+18⇒x=−9,x=1+i,x=1−i

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