Answer: 16
Step-by-step explanation:
12+4= 16
Answer: Increase
Step-by-step explanation:
Answer:
Step-by-step explanation:
This is a third degree polynomial because we are given three roots to multiply together to get it. Even though we only see "2 + i" the conjugate rule tells us that 2 - i MUST also be a root. Thus, the 3 roots are x = -4, x = 2 + i, x = 2 - i.
Setting those up as factors looks like this (keep in mind that the standard form for the imaginary unit in factor form is ALWAYS "x -"):
If x = -4, then the factor is (x + 4)
If x = 2 + i, then the factor is (x - (2 + i)) which simplifies to (x - 2 - i)
If x = 2 - i, then the factor is (x - (2 - i)) which simplifies to (x - 2 + i)
Now we can FOIL all three of those together, starting with the 2 imaginary factors first (it's just easier that way!):
(x - 2 - i)(x - 2 + i) = ![x^2-2x+ix-2x+4-2i-ix+2i-i^2](https://tex.z-dn.net/?f=x%5E2-2x%2Bix-2x%2B4-2i-ix%2B2i-i%5E2)
Combining like terms and canceling out the things that cancel out leaves us with:
![x^2-4x+4-i^2](https://tex.z-dn.net/?f=x%5E2-4x%2B4-i%5E2)
Remembr that
, so we can rewrite that as
and
![x^2-4x+4+1=x^2-4x+5](https://tex.z-dn.net/?f=x%5E2-4x%2B4%2B1%3Dx%5E2-4x%2B5)
That's the product of the 2 imaginary factors. Now we need to FOIL in the real factor:
![(x+4)(x^2-4x+5)](https://tex.z-dn.net/?f=%28x%2B4%29%28x%5E2-4x%2B5%29)
That product is
![x^3-4x^2+5x+4x^2-16x+20](https://tex.z-dn.net/?f=x%5E3-4x%5E2%2B5x%2B4x%5E2-16x%2B20)
which simplifies down to
![x^3-11x+20](https://tex.z-dn.net/?f=x%5E3-11x%2B20)
And there you go!