Answer:
f(x) = x⁴ + 2x³ - 3x² - 4x + 4
Explanation:
we have 3 turnarounds and no negative y values (so, it should be a function based on x⁴), 2 of these turnarounds are also directly on the x-axis (y=0).
f(0) = 4
f(1) = 0
f(-2) = 0
an equation based on x⁴ would have 4 interceptions with the x-axis. since we have only 2 here, then there must be 2 pairs of equal interceptions.
based on the known x-interceptions (x=1, x=-2) we can start with the approach
(x-1)(x+2)
this world give us a quadratic equation (based on x²).
the trick is now to get this to x⁴ without adding additional x-interceptions. that is done by simply "reusing" the ones we already used.
(x-1)(x+2)(x-1)(x+2) = (x-1)²(x+2)²
(x-1)²(x+2)² = (x² - 2x + 1)(x² + 4x + 4) =
x⁴ - 2x³ + x² + 4x³ - 8x² + 4x + 4x² - 8x + 4 =
x⁴ + 2x³ - 3x² - 4x + 4