Find a polynomial of degree n that has only the given zero(s). x = 0, Square root of −6, Square root of 6; n=5

Question

(There are many correct answers)

Answer 1

(x-0)(x-sqr6)(x+sqr6)(x-sqr(-6)(x+sqr(-6)) the last 4 factors come in conjugate pairs

if x-sqr(-6) is a factor, then so is its conjugate x+sqr(-6)

just change the sign of the constant term

if x-sqr6 is a factor, so is x+sqr6

if +sq(-6) is a zero, then x-sqr(-6) is a factor

if +sqr6 is a zero, then x-sqr6 is a factor

if 0 is a zero then x-0 or just x is a factor

= x(x^2-6)(x^2+6) use the general formula (a+b)(a-b) = a^2 - b^2, where a=x^2, b=6

=x(x^4 -36)

= x^5 -36x is the most simple 5 degree polynomial with the given zeros

other answers are any constant times x^5-36x.

let a = any constant. then

a(x^5 -36x)

=ax^5 -36ax where a = any real number,

such as for example a=6,

then the polynomial with the given zeros is 6x^5 - 216x

if a = 1/6, then the polynomial is (1/6)x^5 -6x

if a=1/3, then (1/3)x^5 -12x

if a=3, then 3x^5 - 108x