Question

Write the equation of a cubic function that has zeroes at -2, 3, and 2/5? The function also has a y-intercept of 6?

Answer 1

`color(blue)(y=5/2x^3-7/2x^2-14x+6)`

Explanation:

If `alpha , beta, gamma` are the roots to a cubic function.

Then:

`a(x-alpha)(x-beta)(x-gamma)=0`

Where `bba` is a multiplier.

From given roots:

`a(x+2)(x-3)(x-2/5)=0`

Expanding:

`ax^3-a7/5x^2-a28/5x+a12/5=0`

y intercept is 6:

`a12/5=6`

`12a=30`

`a=30/12=5/2`

`color(blue)(y=5/2x^3-7/2x^2-14x+6)`

GRAPH:

Answer 2

`y = frac{5}{2}x^{3} - frac{7}{2}x^{2} - 14x + 6`

Explanation:

Since `-2`, `3`, and `frac{2}{5}` are all zeroes of the function, we can use the zero product property to find three of the factors:

`(x+2)(x-3)(x-frac{2}{5}) = 0`

Since we also know that `6` is a y-intercept of the function (`y = 6` when `x = 0`), we can find the final factor, some constant (`c`), by solving the following equation for `c`:

`(c)((0)+2)((0)-3)((0)-frac{2}{5}) = 6`
`(c)(2)(-3)(-frac{2}{5}) = 6`
`frac{12c}{5} = 6`
`12c = 30`
`c = frac{5}{2}`

This means that:

`y = (frac{5}{2})(x+2)(x-3)(x-frac{2}{5})`

Which can also be expressed as:

`y = frac{5}{2}x^{3} - frac{7}{2}x^{2} - 14x + 6`