+157 What is the value of the leading coefficient a, if the polynomial function P(x) = a(x + b)^2(x − c) has a multiplicity of 2 at the point (−1, 0) and also passes through the points (7, 0) and (0, −14)?
+23252 If a function has a root at the value x = a, it will have a factor of x - a.
If a function has a root at the point (-1,0), the x-value is -1, so the factor is x - -1 = x + 1.
If it has two roots at that point (multiplicity two), it will have the factor x + 1 twice, or (x + 1)2.
If it passes through the point (7, 0), x = 7, so it has a factor of x - 7.
Thus the function is y = a(x + 1)2(x - 7)
Since it passes through the point (0,-14), replace y with -14 and x with 0.
---> -14 = a(0 + 1)2(0 - 7) ---> -14 = a(1)2(-7) ---> -14 = a(-7) ---> a = 2
So the function is: y = 2(x + 1)2(x - 7)
