Question

Which of these equations has a graph that is tangent to the x-axis at one of its intercepts? Choose all the polynomials that have this property.

y = x^2 + 6x + 9
y = –x^3
y = (x + 2)(x + 6)^2
y = (x – 5)(x + 3)(x + 1)(x – 5)
y = x^4

Answer 1

An equation is tangent to the x-axis when it has an even number of a certain root. A root is a value of x that results to a value of y = 0.
The first equation has roots -3, and -3. The graph will be tangent to the x-axis along with third, the fourth, and the fifth equation by the same condition that they have roots with even multiplicity (repeated in an even number of ways).