Question

Find a polynomial function that has the given zeros:
(2,-6)

Answer 1

So, you want a polynomial such that when x=2,-6 y =0

Answer 2

The easiest way to do this is the following general method:
if f(a) = 0, then the polynomial must contain some factor (x-a)

Answer 3

That may be a little abstract, so let's do the example

Answer 4

so, f(2) and f(-6) equal 0

Answer 5

therefore a possible polynomial could be:
\[(x-2)(x+6)\]

Answer 6

as you can see, when you substitute in 2 or -6 for x, one of the 2 terms will become 0

Answer 7

Now let's foil out the polynomial

Answer 8

\[x^2+4x-12=y\]

Answer 9

Note that this is only one possible polynomial, we could have had:
\[2(x-2)(x+6)\]
which would still be 0 at 2 and -6

Answer 10

however, this would be equal to:
\[2x^2+8x-24 = y\]

Answer 11

which is also a valid solution

Answer 12

So when we put it in a polynomial function it has to be the opposite what was given?

Answer 13

What about (0, 12) ?