Solve for x x^4+15x^3+40x^2-39x+55=0

Problem

x4+15*x3+40*x2−39*x+55=0

Solution

1. Identify the polynomial equation as a quartic equation of the form ƒ(x)=0

2. Apply the Rational Root Theorem to test potential rational roots, which must be factors of the constant term 55 divided by factors of the leading coefficient 1 The possible roots are ±1,±5,±11,±55

3. Test the value x=−5 using synthetic division or direct substitution.

(−5)4+15*(−5)3+40*(−5)2−39*(−5)+55

625−1875+1000+195+55=0

4. Divide the polynomial by (x+5) to find the depressed cubic equation.

(x4+15*x3+40*x2−39*x+55)÷(x+5)=x3+10*x2−10*x+11

5. Test the value x=−11 in the cubic equation x3+10*x2−10*x+11=0

(−11)3+10*(−11)2−10*(−11)+11

−1331+1210+110+11=0

6. Divide the cubic polynomial by (x+11) to find the remaining quadratic factor.

(x3+10*x2−10*x+11)÷(x+11)=x2−x+1

7. Solve the quadratic equation x2−x+1=0 using the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a)

x=(−(−1)±√(,(−1)2−4*(1)*(1)))/(2*(1))

x=(1±√(,1−4))/2

x=(1±√(,−3))/2

x=(1±i√(,3))/2

Final Answer

x=−5,−11,(1±i√(,3))/2

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