find real and complex zeros for f(x) x^3+5x^2+17x+13

Question

f(x)=x³+5x²+17x+13  I have to find the real and complex zeros of the following function?

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.

Rational Root Theorem, if a rational number in simplest form p/q is a root of the polynomial equation a_nxⁿ + a_n – 1x^{n – 1} + ... + a₁x + a₀ = 0, then p is a factor of a₀ and q is a factor if a_n.

f( x ) =  x³+ 5x²+17x+13 = 0

If p/q is a rational zero, then p  is a factor of 13 and q  is a factor of1.

The possible values of p  are   ± 1, and  ± 13, .

The possible values for q  are ± 1.

So, p /q  = ± 1,   ± 13,

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

p/q	1	5	17	13
1	1	6	23	36
13	1	18	251	3276
-1	1	4	13	0

Since f(-1)   =   0,   x   = -1 is a zero. The depressed polynomial is   x²+4x+13 = 0

Since the depressed polynomial of this zero, x²-2x+5, is quadratic, use the Quadratic Formula to find the roots of the related quadratic equation

Substitute a  = 1 , b = 4 , c = 13

Substitute i ^2 = -1.

x = -2±3i

The function has one real zero at x = -1 and two imaginary zeros at x = -2 + 3i and x = -2 - 3i.