Writing equation of a polynomial function
posted by Lisa on .
Can you help me with the following questions, I don't understand. thanks!
Write the equation of a polynomial function with the given characteristics. Use a leading coefficient of 1 or 1 and make the degree of the function as small as possible.
#1. Crosses the xaxis at 4, 0, and 1; lies above the xaxis between 4 and 0; lies below the xaxis between 0 and 1.
#2.Touches the xaxis at 0 and crosses the xaxis at 4; lies below the xaxis between 0 and 4

for #1
xintercepts of a function are solutions to the corresponding equation
since there are solutions of 4,0,and 1
the function must have been f(x) = (x+4)x0)(x1)
or
f(x) = x(x1)(x+4) . You can expand it or leave it as is
for #2
if a graph "touches" the xaxis then there must have been a double root, such as (x2)(x2)
so this one clearly must have been
f(x) = (x0)(x0)(x4)
= x^2(x4)
In both cases we need a cubic for 3 roots. If the x^3 is positive, then the cubic "rises" into the first quadrant and "drops" into the third quadrant.
Since this was the case for both functions there would be a +1 understood in front of the factors,
had it been reversed, a 1 in front of the factors would have achieved this