Solve for x x^4-x^3+4x^2-16x-192=0

Problem

x4−x3+4*x2−16*x−192=0

Solution

1. Test for rational roots using the Rational Root Theorem on the constant term −192 Testing x=4

4−4+4*(4)−16*(4)−192

256−64+64−64−192=0

Since x=4 is a root, (x−4) is a factor.

2. Divide the polynomial by (x−4) using synthetic division or long division to find the remaining cubic factor:

(x4−x3+4*x2−16*x−192)÷(x−4)=x3+3*x2+16*x+48

3. Factor the cubic expression x3+3*x2+16*x+48 by grouping:

x2*(x+3)+16*(x+3)

(x2+16)*(x+3)

4. Identify all factors of the original equation:

(x−4)*(x+3)*(x2+16)=0

5. Solve for x by setting each factor equal to zero:

x−4=0⇒x=4

x+3=0⇒x=−3

x2+16=0⇒x2=−16⇒x=±4*i

Final Answer

x=4,−3,4*i,−4*i

Want more problems? Check here!