No worries! I'll guide you through the steps to find the function in both completely factored form and polynomial form.
(a) To write the function in completely factored form, we use the zeros of the function. Given that the zeros are -2, 2+2√2i, and 2-2√2i, we can start by setting up factors for each of these zeros.
The factor for -2 can be written as (x - (-2)), which simplifies to (x + 2).
Now, let's focus on the complex zeros 2+2√2i and 2-2√2i. Complex zeros always come in conjugate pairs, which means if we have one, we'll have the other. The complex conjugate of 2+2√2i is 2-2√2i, and vice versa. So the factors for the complex zeros are (x - (2 + 2√2i)) and (x - (2 - 2√2i)).
Next, we multiply these factors together to get the completely factored form. Here's the expression:
f(x) = (x + 2)(x - (2 + 2√2i))(x - (2 - 2√2i))
(b) To write the function in polynomial form, we need to expand and simplify the expression obtained in part (a) by multiplying all the factors:
f(x) = (x + 2)(x - (2 + 2√2i))(x - (2 - 2√2i))
To simplify this expression, we need to use the conjugate property of complex numbers. When we multiply a complex number by its conjugate, we eliminate the imaginary part.
Let's multiply the conjugates first:
(x - (2 + 2√2i))(x - (2 - 2√2i))
Using the formula (a + b)(a - b) = a^2 - b^2, we can simplify it to:
(x^2 - (2 + 2√2i)^2)
Expanding the squared term:
(x^2 - (2^2 + 2 * 2 * 2√2i + (2√2i)^2))
Simplifying and using the fact that (2√2i)^2 = 4(√2i)^2 = 4 * 2i = 8i:
(x^2 - (4 + 8√2i - 8i))
(x^2 - 4 - 8√2i + 8i)
Now we can multiply all the remaining factors:
f(x) = (x + 2)(x^2 - 4 - 8√2i + 8i)
Expand and simplify further:
f(x) = x^3 + 2x^2 - 4x - 8√2i - 8i - 16 - 32√2i + 32i
Finally, simplify the terms:
f(x) = x^3 + 2x^2 - 4x - 16 + (24 - 8√2i + 32i - 32√2i)
f(x) = x^3 + 2x^2 - 4x - 16 + (-8√2i + 32i - 32√2i + 24)
Combining like terms:
f(x) = x^3 + 2x^2 - 4x - 8√2i - 32√2i + 32i + 24 - 16
f(x) = x^3 + 2x^2 - 4x - (8 + 32)√2i + 32i + 8
f(x) = x^3 + 2x^2 - 4x - 40√2i + 32i + 8
So, the polynomial form of the function is:
f(x) = x^3 + 2x^2 - 4x - 40√2i + 32i + 8
I hope this explanation helps you understand how to find the completely factored form and polynomial form of a given polynomial function with specific zeros and solution points!