Question

Write a polynomial function in standard form for the set of zeros. Show your step-by-step work.
x=2, i, −i

Answer 1

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