Given the following information about a polynomial function, find the function.

Question

It has a double zero at x=2 and zeroes at x=−1 and x=3. It’s y-intercept is (0,−4).

f(x)=

Answer 1

Each of the zeros are going to come from a factor. In general, if you have a zero of x = c, then it comes from a factor of (x - c).

So since we have zeros of x = 2, -1, and 3, then we must have factors of (x - 2)(x + 1)(x - 3). Since x = 2 is a double zero, this means it occurs twice, so we can write that as (x - 2)². Putting it all together, our function so far should look like:

y = (x - 2)²(x + 1)(x - 3)

This would be sufficient, but we have one last criteria, and that is a y - intercept at (0,-4). Plugging in 0 for x and -4 for y, we can find the appropriate coefficient to our function, I'll call it 'a'.

y = a(x - 2)²(x + 1)(x - 3)

-4 =a(0 - 2)²(0 + 1)(0 - 3)

-4 = a(4)(1)(-3)

-4 = -12a

1/3 = a

So our a value is 1/3 and the complete function should be:

y = 1/3(x - 2)²(x + 1)(x - 3)

The only other thing you might be required to do here is to multiply out the whole function to put it in standard form. This was not asked for in the question, so I will just leave the function in factored form.