Question

Find the roots of the polynomial equation -x^3 + 5x^2 - 11x + 55 = 0.

Answer 1

Given polynomial equation -x^3+5x^2-11x+55 = 0

By synthatic division

 

Answer 2

  

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

- x³+ 5x²- 11x + 55 = 0

If p/q is a rational zero, then p  is a factor of 55 and q  is a factor of 1.

The possible values of p  are   ± 1,   ± 5, ± 11 and  ± 55,.

The possible values for q are ± 1.

So, p/q = ± 1,   ± 5, ± 11,± 55.

Make a table for the synthetic division and test possible  zeros.

p/q	-1	5	-11	55
1	-1	4	-5	50
-1	-1	6	-17	72
-5	-1	10	-61	360
5	-1	0	-11	0

Since f (5) = 0,   x =  5 is a zero. The depressed polynomial is   - x ²- 11 = 0

Since the depressed polynomial of this zero, - x ²- 11 , is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

Substitute a  = -1 , b  = 0 , c  = -11

Roots of the polynomial are one real root at x  = 5 and two imaginary roots at x = -i√11 and

x =  +i√11.