Write a polynomial function in standard form for the set of zeros. Show your step-by-step work.
x=2, i, −i
To find the polynomial function in standard form for the given set of zeros, we need to first identify the factors corresponding to each zero.
1) x = 2
Since x = 2 is a zero, we know that (x - 2) is a factor of the polynomial.
2) x = i
Since x = i is a zero, we know that (x - i) is a factor of the polynomial.
3) x = -i
Since x = -i is a zero, we know that (x + i) is a factor of the polynomial.
Next, we multiply these factors together to find the polynomial function:
(x - 2)(x - i)(x + i)
Using the difference of squares identity (a^2 - b^2 = (a + b)(a - b)), we can simplify this expression:
(x - 2)(x^2 + i^2)
(x - 2)(x^2 + (-1))
(x - 2)(x^2 - 1)
Now, we multiply the remaining factors:
(x - 2)(x + 1)(x - 1)
Expanding this expression:
(x^2 - x + 2x - 2)(x - 1)
(x^2 + x - 2)(x - 1)
Finally, we distribute the (x - 1) term:
x^3 - x^2 + 2x^2 - x - 2x + 2
Combining like terms:
x^3 + x^2 - 3x + 2
Therefore, the polynomial function in standard form for the given set of zeros is:
f(x) = x^3 + x^2 - 3x + 2