Final answer:
The statement is false; the polynomial P(x) has roots at x=6 and x=-5, and since x=-3 is not a root, the graph of y=P(x) is not tangent to the x-axis at (-3,0).
Step-by-step explanation:
The statement given is that if P(x) = (x - 6)⁶(x + 5)⁷, then the graph of the polynomial y=P(x) is tangent to the x-axis at (-3,0), which is false.
To determine where a polynomial is tangent to the x-axis, we must look at its roots and their corresponding multiplicities. In this polynomial, the roots are x=6 and x=-5, and both roots have multiplicities greater than 1, which indicates that the graph will touch and turn at these x-values rather than crossing the x-axis.
Since the root x=-3 is not a part of the given polynomial equation, P(x) cannot have a value of zero at x=-3.
Therefore, the graph of y=P(x) will not touch or intersect the x-axis at (-3,0), hence it cannot be tangent to the x-axis at that point.