Answer:
![f(x)=x^5-6x^4+14x^3-24x^2+40x](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E5-6x%5E4%2B14x%5E3-24x%5E2%2B40x)
Step-by-step explanation:
<u>Given information</u>:
- Polynomial function with real coefficients.
- Zeros: 0, 2i and (3+i).
For any complex number
, the complex conjugate of the number is defined as
.
If f(z) is a polynomial with real coefficients, and z₁ is a root of f(z)=0, then its complex conjugate z₁* is also a root of f(z)=0.
Therefore, if f(x) is a polynomial with real coefficients, and 2i is a root of f(x)=0, then its complex conjugate -2i is also a root of f(x)=0.
Similarly, if (3+i) is a root of f(x)=0, then its complex conjugate (3-i) is also a root of f(x)=0.
Therefore, the polynomial in factored form is:
![f(x)=ax(x-2i)(x-(-2i))(x-(3+i))(x-(3-i))](https://tex.z-dn.net/?f=f%28x%29%3Dax%28x-2i%29%28x-%28-2i%29%29%28x-%283%2Bi%29%29%28x-%283-i%29%29)
![f(x)=ax(x-2i)(x+2i)(x-3-i)(x-3+i)](https://tex.z-dn.net/?f=f%28x%29%3Dax%28x-2i%29%28x%2B2i%29%28x-3-i%29%28x-3%2Bi%29)
As we have not been given a leading coefficient, assume a = 1:
![f(x)=x(x-2i)(x+2i)(x-3-i)(x-3+i)](https://tex.z-dn.net/?f=f%28x%29%3Dx%28x-2i%29%28x%2B2i%29%28x-3-i%29%28x-3%2Bi%29)
Expand the polynomial:
![f(x)=x(x^2+2ix-2ix-4i^2)(x^2-3x+xi-3x+9-3i-xi+3i-i^2)](https://tex.z-dn.net/?f=f%28x%29%3Dx%28x%5E2%2B2ix-2ix-4i%5E2%29%28x%5E2-3x%2Bxi-3x%2B9-3i-xi%2B3i-i%5E2%29)
![f(x)=x(x^2-4i^2)(x^2-6x+9-i^2)](https://tex.z-dn.net/?f=f%28x%29%3Dx%28x%5E2-4i%5E2%29%28x%5E2-6x%2B9-i%5E2%29)
![f(x)=x(x^2-4(-1))(x^2-6x+9-(-1))](https://tex.z-dn.net/?f=f%28x%29%3Dx%28x%5E2-4%28-1%29%29%28x%5E2-6x%2B9-%28-1%29%29)
![f(x)=x(x^2+4)(x^2-6x+10)](https://tex.z-dn.net/?f=f%28x%29%3Dx%28x%5E2%2B4%29%28x%5E2-6x%2B10%29)
![f(x)=(x^3+4x)(x^2-6x+10)](https://tex.z-dn.net/?f=f%28x%29%3D%28x%5E3%2B4x%29%28x%5E2-6x%2B10%29)
![f(x)=x^5-6x^4+10x^3+4x^3-24x^2+40x](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E5-6x%5E4%2B10x%5E3%2B4x%5E3-24x%5E2%2B40x)
![f(x)=x^5-6x^4+14x^3-24x^2+40x](https://tex.z-dn.net/?f=f%28x%29%3Dx%5E5-6x%5E4%2B14x%5E3-24x%5E2%2B40x)
9514 1404 393
Answer:
∠4 = 100°
Step-by-step explanation:
This has nothing to do with parallel lines.
Angles 1 and 4 are "vertical angles", hence congruent.
∠1 = ∠4 = 100°
_____
Vertical angles are on opposites sides of the intersection where lines cross.
Answer:
Step-by-step explanation:
x= -1 and -3
What when multiplied is +3 but when added is 4? 3 and 1. You then get (x+3)(x+1) is the first part of the answer. If you want to find x, set x+3 to 0. x+3=0. Then subtract 3 from both sides to get x by itself. x= -3. Do the same with the x+1.
Ok, so to increase 200 by 18%, you want to know how much is 1% of 200. So, in order to figure this out, you want to divide 200 by 100, to get 2. So, you know that 2 is 1% of 200. Now, to increase 200 b 18%, you will want to multiply 18 by 2, to get 26. So, the final answer is 236.