How do you write a polynomial in standard form given the zeros x=-6, 2, and 5?
A polynomial with zeros
You are given the following information about the polynomial: zeros.
Definition of zeros: If x = zero value, the polynomial becomes zero. i.e. If you plug in -6, 2, or 5 to x, this polynomial you are trying to find becomes zero.
For the polynomial to become zero at let's say x = 1, the polynomial needs to contain the following term:
This looks too simple, and polynomials are usually bigger. They are usually quadratic, cubic, quartic, etc.
However, those messy polynomials all have
Because we have 3 zeros, we write the same kind of equation for each zero.
You multiply all of them to get the following equation.
If you are wondering why we multiply them, I'll give you a hint. Zero multiplied by any number is zero. So if one of the three equations above becomes zero, does it matter what values the other two equations give?
Expand the equation:
And there's your answer!
One extra note: a more complete answer would be
The polynomial equation with roots a, b and c is
The cubic equation with roots a, b and c is
Here (a, b, c) = (6, 2, 5).
a+b+c+ 13, bc+ca+ab=52 abd abc=60