This is the second part of a two part question for an online class. It gave me the degree and the zeros and I had to give the factored form. I got that part right, but I need to know how to get the expanded form from the factored form. I have several more questions like this and the knowledge of how to do them would be appreciated.
Just for the record the "i"s are imaginary numbers.
Factored= 10(x-2)(x+2)(x-3i)(x+3i)
notice you have 2 groupings of 'difference of squares', I would do those first
10(x-2)(x+2)(x-3i)(x+3i)
= 10(x^2 - 4)(x^2 + 9)
now just slug it out ...
= 10(x^4 + 9x^2 - 4x^2 - 36)
= 10x^4 + 50x^2 - 360
let's use Wolfram to check:
http://www.wolframalpha.com/input/?i=factor+10x%5E4+%2B+50x%5E2+-+360
Yup, works
I would look for special groupings of factors like above. If you don't see any, just grind it out, one multiplication at a time.
Always simplify before doing the next multiplication.
Thanks for the help I really appreciate it.
To expand the factored form, you need to multiply out all the factors. In this case, you have four factors: (x-2), (x+2), (x-3i), and (x+3i).
To expand the factored form, you need to multiply these factors together using the distributive property.
Let's start by expanding the first two factors, (x-2) and (x+2):
(x-2)(x+2) = x * x + x * 2 - 2 * x - 2 * 2
= x^2 + 2x - 2x - 4
= x^2 - 4
Next, let's expand the third and fourth factors, (x-3i) and (x+3i):
(x-3i)(x+3i) = x * x + x * 3i - 3i * x - 3i * 3i
= x^2 + 3ix - 3ix - 9i^2
= x^2 - 9i^2 (Note: i^2 = -1)
= x^2 + 9
Now, we need to multiply the two expanded expressions:
(x^2 - 4)(x^2 + 9) = x^2 * x^2 + x^2 * 9 - 4 * x^2 - 4 * 9
= x^4 + 9x^2 - 4x^2 - 36
= x^4 + 5x^2 - 36
So, the expanded form of the given factored form 10(x-2)(x+2)(x-3i)(x+3i) is x^4 + 5x^2 - 36.