Question

What is the polynomial equation for the points passing through (-2,0), (0,0), (1,-4), (2,0)?

Answer 1

`f(x) = 4/3 (x^3-4x)`

Explanation:

First let defined a polynomial function in factor form as follow

`f(x) = a(x-b)(x-c)(x-d).....(x-z)`,

where `a` is a none zero leading coefficient and

`b, c, d.....z` are the`x` intercepts, of the function, which meant
`x= b, x= c, x= d...... x= z`

We are given the following x- intercepts

`x = -2 , y = 0`
`x= , y = 0`
`x= 2, y= 0`

And the value of
`x= 1, y = -4`

We can re-write the x-intercepts in the factor forms like this

`f(x) = (x)(x+2)(x-2)`

the general form is

`f(x) = a(x)(x-2)(x+2)`

`f(x) = a(x(x^2-4)`

`f(x) = a(x&3 - 4x)`

Since `x= 1, y = -4` , we can substitute that into the general form to solve for `a`

`f(1) = a(1^3 -4(1))`
`-4 = a(1-4)`
`-4 = -3a`
`a= 4/3`

So the general polynomial function that passing through the points
`(0,0),(2, 0), (-2,0), (1,-4)` is

`f(x) = 4/3 (x^3-4x)`

`f(x)= 4/3 x^3 - (16x) /3`