Question

The number of zeroes that polynomial f(x) = (x – 2)^12 + 4 can have is: *

Answer 1

Answer:

The number of zero is equal to degree of the polynomials

Therefore the no of zero is 12

Answer 2

Answer:

Well, this polynomial can have zero real roots because both the terms will be absolutely positive if x is real, however if x is allowed to be complex then, there'll surely be 12 roots of this polynomial equation.

In general, the polynomial will have 12 roots as it is the degree of the polynomial, this can be verified by expanding the first bracket by using binomial theorem.

To find these roots suppose x = z

Now,

to find, all such z such that

(z-2)^12 = -4 = (4)(cos (0+2mπ) +i sin (2mπ+0)) =4 cis (2mπ)

where,

z= 2+ (2)^2/12 cis(2mπ/12)

or,

x= 2 + 2^1/6 (cos mπ/6 + i sinmπ/6)

such that m takes the values

0,1,2,.....,11

so that there will be 12 corresponding roots to the equation.

Hope this helps you !