How do I find the Equation from two+ roots and vice-versa?

Question

How do I get the equation if there's more than two roots? And what is the method to graph it?

Also, how to determine the roots from polynomial, which has more than two roots? (like x^3-2x^2+x-2).

I need steps and please try to explain it more simpler. Ask me if you don't understand anything that I asked about.

Answer 1

No matter how many roots there are, you can try to use the rational root theorem, if all of the coefficients are integers. If they are fractions, then you can multiply by the least common multiple of the denominators to get integer coefficients.

Assuming they are integers, suppose the polynomial has the form
an x^n + an-1 x^n-1 + ... + a2 x^2 + a1 x + a0.
The rational root theorem says that any rational root of this polynomial has the form p/q, where p is a factor of a0, and q is a factor of an.

For the polynomial you just gave, x^3 - 2x^2 + x - 2, the rational root theorem implies that any rational root must be a factor of 2 over a factor of 1. So try x=1 and x=2. You'll find that 1 is not a root, but 2 is.

Now, divide the polynomial by x-2. You'll see that you can factor x-2 out as follows:
x^3 - 2x^2 + x - 2 = (x-2)x^2 + (x-2) = (x-2)(x^2+1).
Note: in this case, it could be seen that x-2 could be factored out, which would reveal that it's a root, but that's not always the case. You can always try long division or synthetic division if there's no obvious factorization.

Now you can get the roots of x^2+1 using the quadratic formula.

Answer 2

assume the polynomial takes the form: p(x) = ax^2 + bx + c, and so p'(x) = 2ax + b. p(x) = 0 ? x = [-b ± ?(b^2 - 4ac)] / (2a) p'(x) = 0 ? x = -b / (2a) Now, all it incredibly is left to instruct is that ?z??, [-b - ?(b^2 - 4ac)] / (2a) < -b / (2a) < [-b + ?(b^2 - 4ac)] / (2a) merely carry out a little algebra and simplify and it incredibly is going to be a chunk of cake. Crap, I merely found out I purely did it for 2d degree polynomials, i'm going to do it for all n interior the morning.