Find the Roots (Zeros) f(x)=x^3+9x^2-25x-33

Problem

ƒ(x)=x3+9*x2−25*x−33

Solution

1. Identify potential rational roots using the Rational Root Theorem, which suggests testing factors of the constant term −33 (specifically ±1,±3,±11,±33.

2. Test x=−1 by substituting it into the function: ƒ*(−1)=(−1)3+9*(−1)2−25*(−1)−33=−1+9+25−33=0 Since ƒ*(−1)=0 (x+1) is a factor.

3. Divide the polynomial x3+9*x2−25*x−33 by (x+1) using synthetic division or long division to find the remaining quadratic factor.

4. Result of the division is the quadratic expression x2+8*x−33

5. Factor the quadratic expression x2+8*x−33 by finding two numbers that multiply to −33 and add to 8 which are 11 and −3

6. Write the fully factored form of the function: ƒ(x)=(x+1)*(x+11)*(x−3)

7. Solve for the roots by setting each factor equal to zero: x+1=0 x+11=0 and x−3=0

Final Answer

x=−1,−11,3

Want more problems? Check here!