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 (–3, 0) and also passes through the points (2, 0) and (0, 36)?

Guest Mar 6, 2018

#1**+1 **

If we have a root at (-3,0) with a multiplicity of 2 and the point (2,0) is aslo on the graph.....we have three roots.....but this is a polynomial of degree 2, so it can't possibly have three roots.....did you make a mistake ??

CPhill
Mar 6, 2018

#2**+1 **

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

P(x) = a(x + b)(x + b)(x - c) Multiplied out.....

P(x) = ax^{3} - acx^{2} + 2abx^{2} - 2abcx + ab^{2}x - ab^{2}c So P(x) does have a degree of 3 .

P(x) has multiplicity of 2 at (-3, 0) and passes through (2, 0) so...

P(x) = a(x + 3)^{2}(x - 2)

P(x) passes through (0, 36) so P(0) = 36

P(0) = a(0 + 3)^{2}(0 - 2)

36 = a(0 + 3)^{2}(0 - 2)

36 = a(9)(-2)

36 = -18a

a = -2

Here's a graph: https://www.desmos.com/calculator/ic91u5dxbl

hectictar
Mar 7, 2018