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

B. 4 imaginary; 1 real

Step-by-step 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.

a line is an defined derm because it

a)contains an infinite number of points b) can be used to create other geometric shapes c)describes something that does not have width or depth d) is a term that does not have a formal definition. a line is an undefined term because it: contains an infinite number of points.

step-by-step explanation:

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