Solve the polynomial equation.

Question

Solve the polynomial equation.In order to obtain the first root use synthetic division to test the possible rational roots.

 

3x^4+26x^3+91x^2+128x+52=0                ^{(SHOW ALL THE WORK)}

 

 

 

Answer 1

According to the rational root theorem, the possible roots are the ratio of the last terms factors to the factors of the first term's coefficients of the polynomial.

 

 

Factors of 52:  1, 2, 4, 13, 26, 52

Factors of 3:  1, 3

 

 

Possible roots = ±1, ±2, ±4, ±13, ±26, ±52, ±1/3, ±2/3, ±4/3, ±13/3, ±26/3, ±52/3

 

 

Use synthetic division to see which one is a root. We need the remainder to be zero.  Since each term of the equation is positive, the only possible roots is then negative.

 

 

I would start testing these roots:

 

x = -1/3

x = -2/3

x = -4/3

x = -13/3

x = -26/3

x = -52/3

 

 

Once you have your root, put it in factor form.  The quotient that you got will be the other factor in which when you multiply them together, you get the original equation.

 

Try to factor out the larger factor by grouping, FOIL, or taking out the GCF.  Otherwise, use synthetic division one last time.  You keep doing this until you obtain a quadratic factor. 

 

 

Once you have all your factors in simplest form, it will be easier for you to solve for the solutions.