Question

find the real and complex zeros of the following function f(x)=x^3-5x^2+8x-6

Answer 1

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.

f (x ) = x³- 5x² + 8x - 6

If p/q is a rational zero, then p is a factor of 6 and q is a factor of 1.

The possible values of p are   ± 1, ± 2, ± 3 and   ± 6.

The possible values for q are ± 1.

So, p/q = ± 1, ± 2, ± 3 and   ± 6.

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

p/q	1	-5	8	-6
-1	1	-6	14	-20
1	1	-4	4	-2
2	1	-3	2	-2
-2	1	-7	22	-50
3	1	-2	2	0

Since f (3)  = 0, x  =  3 is a zero. The depressed polynomial is   x²- 2x + 2 = 0

Since the depressed polynomial of this zero, x²- 2x + 2, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

x²- 2x + 2 = 0

Roots are

a = 1, b = - 2 ,c = 2

The functin has one real solution at x  = 3 and two complex solutions are at x  = 1 + i and x  = 1- i.