writing the function given zeros

Question

I have to write the simplest function given 3 and 2+1. i need steps and im so confused.

Answer 1

If the roots are 3, 2, and 1, then we can apply the Factor Theorem, which says that for any zero k that a function has, the corresponding factor is x-k.

That means that the factors of this function are x-3, x-2, and x-1, so the simplest form of the function is f(X)=(x-3)(x-2)(x-1). Just multiply those out, then.

Now if the given roots are 3 and 2+i, then we have imaginary roots. Provided the polynomial has real coefficients (that's the norm for high school math), then 2-i is also a root, since imaginary roots come in conjugate pairs in such situations.

From the Factor Theorem, we know that x-3 will be one of the factors. For the others, though, we can utilize a little "trick." The roots 2+i and 2-i are solutions to a quadratic equation. We can find that equation by working backward from either of the roots:

x=2+i

x-2=i

(x-2)²=i²

x²-4x+4=-1

x²-4x+5=0

This means that the product of (x-(2+i)) and (x-(2-i)) is equal to x²-4x+5. We then take that and multiply it by our other factor to yield our polynomial function: f(x)=(x-3)(x²-4x+5).