Question

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)?
2

Answer 1

If the polynomial function $P(x) = a(x + b)^2(x-c)$ has a multiplicity of 2 at the point (−1, 0) then the factor (x-(-1))=(x+1) twice enters the polynomial representation. So, you have known one part of the left side of the polynomial function that is $P(x) = a(x + 1)^2(x-c)$.

If polynomial function passes through the point (7,0), then $0 = a(7+ 1)^2(7-c)$ and since $a\neq 0$ you have that c=7.

At last, if polynomial function passes through the point (0,-14), then $-14 = a(0+ 1)^2(0-7)$ and $-14=-7a$, so a=2.
Answer: a=2.