Question

Find a cubic function with the given zeros. You must show your work for credit.

-2, 3, -3
please explain how to do this!

Answer 1

As a special bonus, I'll tell you how to find equations of any degree with given zeros :-)

If you have a polynomial \(P(x)\) with a set of roots \(r_1, r_2, ... r_n\) it can be written as a product of its factors and an arbitrary constant, \(k\): \(k(x-r_1)(x-r_2)...(x-r_n)\). \(k\) doesn't affect the roots, just the vertical scale of the curve. You can just use \(k=1\) if you don't have to fit the curve to both roots and a specific point.

Answer 2

You don't in this problem, but sometimes you'll be told that you have a root with multiplicity (some number). What that means is that you need multiple copies of the binomial for that root. For example, \(y=x^2\) is a parabola with roots at x = 0 and multiplicity 2, so it is \(y = (x-0)^2\) in the form above.

You'll probably need to expand the polynomial (multiply it out) to get full credit, though in my opinion the learning part is knowing how to construct it from the roots.

Answer 3

i kind of get it, but can you write out the problem so i can see it. please..

Answer 4

Okay, I'll do it with a different set of roots. roots are -1, -2, and 3:

\[P(x) = 1(x-(-1))(x-(-2))(x-3) = (x+1)(x+2)(x-3)\]If we expand that,\[P(x) = (x^2+3x+2)(x-3) = x^3 -7x -6\]

Answer 5

The reason this all works is that the roots are just the spots where the function has a value of 0. We write the function as a product of binomials \((x-r_n)\), which means that if any of the binomials evaluates to 0 at a given value of \(x\), the product will be 0. The values of x where the function will be 0 will be exactly the values of x where it has a root, and vice versa.

Answer 6

It should also illustrate why the number of roots is equal to the highest exponent...