Question

How do you find a polynomial function that has zeros x=-2, 4, 7 and degree n=3?

Answer 1

`x^3-9x^2+6x+56`

Explanation:

If a polynomial has zeros: `color(red)(x=-2), color(blue)(x=4), and color(green)(x=7)`
then it has factors
`color(white)("XXX")(x-(color(red)(-2))), (x-color(blue)(4)), and (x-color(green)7)`

Furthermore, if the polynomial has degree `3`, then these `3` factors are the only factors and the polynomial is
`color(white)("XXX")(x-(color(red)(-2)))xx(x-color(blue)4)xx(x-color(green)7)`

#{: (,underline(xx),underline(" | "),underline(x),underline(color(red)(+2))), (,x," | ",x^2,+2x), (,underline(color(blue)(-4)),underline(" | "),underline(-4x),underline(-8)), (," | ",x^2,-2x,-8), (,,,,), (,,,,), (underline(xx),underline(" | "),underline(x^2),underline(-2x),underline(-8)), (x," | ",x^3,-2x^2,-8x), (underline(color(green)(-7)),underline(" | "),underline(-7x^2),underline(+14x),underline(+56)), (,x^3,-9x^2,+6x,56) :}#