Question

What is the value of the leading coefficient a if the polynomial function P(x) = a(x + b)2(x − c) has multiplicity of 2 at the point (−1, 0) and also passes through the points (4, 0) and (0, −12)? (4 points)

A) 2
B) −3
C) 3
D) −12

Answer 1

The given polynomial function is:


`P(x)=a (x+b)^(2)(x-c)`

It is given that the P(x) has a zero at (-1,0) with multiplicity 2 and another zero at (4,0). So, b=1 and c=4 for the above equation. Substituting the values in equation of P(x), we get:


`P(x)=a (x+1)^(2)(x-4)`

P(x) passes through the point (0,-12). Thus P(0) must be 12. Setting P(0) equal to 12, we get:


`P(0)=a (0+1)^(2)(0-4) \\ \\ 12=a(-4) \\ \\ a=-3`

The polynomial thus becomes:


`P(x)=-3 (x+1)^(2)(x-4)`

Therefore, leading coefficient, a , of the polynomial is -3. B option is the correct answer.