Question

Find a cubic function with given zeroes

I don't know how to do this

Answer 1

\[\sqrt{6}, -\sqrt{6}, -3\]

Answer 2

A cubic function can have up to 3 real roots (or "zeroes".) To find a cubic function with given roots a, b, and c, try multiplying together (x-a)(x-b)(x-c). You can confirm that each of the zeroes give you a result of zero by plugging them into these factors. In your example, this would give you:

x -\[\sqrt{6}\]
x+\[\sqrt{6}\]
x+3

Because two of the roots are negative, you would get a double negative which would give you a positive.

Answer 3

@vinnv226 can you help me with the first step because I'm a little confused

Answer 4

Recall that any value times 0 gives you 0, and the only way for two (or more) numbers to multiply together and get 0 is for one of them to be 0.

You mentioned that -3 is a zero of the function. In order for this to be true, one of the factors of the function must be (x- -3) which is (x+3). If (x+3) is not a factor in the equation, then it's impossible to plug in -3 and get 0. The same idea follows with the other zeroes. So, to get the cubic function with the zeros you mentioned, we need to multiply together these three factors:

\[x+3\]
\[x-\sqrt{6}\]
\[x+\sqrt{6}\]