find the complex and real zeros of the following function

Question

need to find the real and complex zeros of f(x) = x^3+2x^2-6x+8

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.

The function f (x ) =  x³ + 2x²- 6x + 8

If p/q is a rational zero, then p is a factor of 8 and q is a factor of 1.

The possible values of p are   ± 1,   ± 2, ± 4 and   ± 8.

The possible values for q  are ± 1.

So, p /q  = ± 1,   ± 2, ± 4 and   ± 8.

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

p/q	1	2	-6	8
1	1	3	-3	5
-1	1	1	-7	15
-2	1	0	-6	20
2	1	4	2	12
3	1	5	9	35
-3	1	-1	-3	17
-4	1	-2	2	0

Since f (-4)   =   0,   x   =   -4 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  = - 4 and two complex solutions are at x  = 1 + i and

x  = 1- i.