Explanation:
a. both, the left and the right end behaviors go to +infinity (that means for positive and for negative x).
so, the strongest and highest exponent of the variable (= the degree of the polynomial) must be an even number. because with an uneven number negative x would go to -infinity, or at least to the opposite infinity than the one for the positive numbers.
and since the variable has an even number exponent, that value is positive. so, "a" must be positive too, so that the curve opens upwards (as we had already for parabolas) and allows for the +infinity limits.
b. (-3, 0), (2, 0), (5, 0)
(2, 0) counts twice, as it is only touching the x-axis. if e lifted the curve up a little bit, this would immediately turn into 2 separate zeroes.
so, we have 4 zeros. that means the degree of the polynomial is 4 (the highest exponent of the variable is 4).
c.
in the factors form we convert every zero into a factor of the function that is equal to 0 for the particular zero point. remember, (2, 0) counts twice.
so, we get
y = a(x + 3)(x - 2)²(x - 5)
when we use (0, -2) in this equation we get
-2 = a(3)(-2)²(-5) = a×3×4×-5 = -60a
a = -2/-60 = 1/30
so, or final equation is
y = (1/30)(x + 3)(x - 2)²(x - 5)