Question

State how many imaginary and real zeros the function has.

f(x) = x5 + 7x4 + 2x3 + 14x2 + x + 7
A. 3 imaginary; 2 real
B. 4 imaginary; 1 real
C. 0 imaginary; 5 real
D. 2 imaginary; 3 real

Answer 1

B. 4 imaginary; 1 real

Explanation:

Given the polynomial:

x^5 + 7*x^4 + 2*x^3 + 14*x^2 + x + 7

it can be reordered as follows

(x^5 + 2*x^3 + x ) + (7*x^4 + 14*x^2 + 7)

Taking greatest common factor at each parenthesis

x*(x^4 + 2*x^2 + 1) + 7*(x^4 + 2*x^2 + 1)

Taking again the greatest common factor

(x + 7)*(x^4 + 2*x^2 + 1)

Replacing x^2 = y in the second parenthesis

(x + 7)*(y^2 + 2*y + 1)

(x + 7)*(y + 1)^2

Coming back to x variable

(x + 7)*(x^2 + 1)^2

There are two options to find the roots

(x + 7) = 0

or

(x^2 + 1)^2 = 0 which is the same that (x^2 + 1) = 0

In the former case, x = -7 is the real root. In the latter, (x^2 + 1) = 0 has no real solution. Therefore, there is only 1 real root in the polynomial.