write a polyoil equation ofleast degree for each set of roots +/-2i,3,-3.
To write a polynomial equation of least degree with the given set of roots, we can use the fact that complex roots always appear in conjugate pairs.
Given roots:
-2i, 2i, 3, -3
To find the polynomial equation, we start by considering the factors that contribute to the roots. For each root, we have:
Root: -2i
Factor: (x - (-2i)) = (x + 2i)
Root: 2i
Factor: (x - 2i)
Root: 3
Factor: (x - 3)
Root: -3
Factor: (x - (-3)) = (x + 3)
To obtain the polynomial equation, we multiply all the factors together:
(x + 2i)(x - 2i)(x - 3)(x + 3)
To simplify this equation, let's start by multiplying the conjugate pairs:
(x^2 - (2i)^2)(x - 3)(x + 3)
Simplifying further:
(x^2 + 4)(x - 3)(x + 3)
Expanding the remaining terms:
(x^2 + 4)(x^2 - 9)
Now we can multiply each term:
x^4 - 9x^2 + 4x^2 - 36
Combining like terms:
x^4 - 5x^2 - 36
Therefore, the polynomial equation of least degree with the given set of roots is:
f(x) = x^4 - 5x^2 - 36