Question

HELP!!!!!!!!!!!!!
write equation of a polynomial function with the given characteristics.
Leading coefficient is 1 or -1
crosses the x axis at -2,3, and 4
touches the x axis at 0
lies above the x axis between -2 and 0

Answer 1

from "touches the x axis at 0", we can see a double root of x=0 at the origin, meaning the curve must touch the x-axis at the origin without crossing.
But it does cross at x = -2,3 and 4

so we would have
y = ±1(x^2)(x+2)(x-3)(x-4)

[think of the x^2 as (x-0)(x-0)]

make a sketch with intercepts as I have indicated, having the curve rise to infinity in the 1st, and to negative infinity in the 3rd quadrants,
for the curve to be above the x-axis between -2 and 0
so the coefficient has to be +1

so y = (x^2)(x+2)(x-3)(x-4)

Answer 2

Can you write an equation of a polynomial function with given characteristics:

one x-intercept
end behaviour of Q2->Q4
y-intercept of (0,2)