Given that (+-2) and i are the roots of a 4th degree polynomical, find the equation.
Please help! Thank you:)
complex roots come in pair, so -i is also a root
f(x) = (x-2)(x+2)(x-i)x+i)
= (x^2-4)(x^2+1)
= x^4 - 3x^2 - 4
Thanks!
To find the equation of a 4th degree polynomial with the given roots (+-2) and i, we can start by writing out the factors of the polynomial.
Since the roots (+-2) are real, they exist in conjugate pairs. So, the factors involving (+-2) are (x - 2) and (x + 2).
The root "i" means that the polynomial has a factor of (x - i) or (x + i).
Now, we can multiply these factors together to obtain the equation of the polynomial:
(x - 2)(x + 2)(x - i)(x + i)
To simplify further, we can use the difference of squares identity: (a^2 - b^2) = (a + b)(a - b).
Applying this to the first two factors, (x - 2)(x + 2), we get (x^2 - 4).
Using the same identity for the last two factors, (x - i)(x + i), we get (x^2 - i^2), which simplifies to (x^2 + 1).
Combining these, the equation of the 4th degree polynomial is:
(x^2 - 4)(x^2 + 1)
Expanding this equation gives us the final answer:
x^4 - 3x^2 - 4
Therefore, the equation of the 4th degree polynomial with roots (+-2) and i is x^4 - 3x^2 - 4.