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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)?

 Mar 6, 2018
 #1
avatar+97967 
+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 ??

 

 

cool cool cool

 Mar 6, 2018
 #2
avatar+7350 
+1

P(x)   =   a(x + b)2(x - c)

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

P(x)   =   ax3 - acx2 + 2abx2 - 2abcx + ab2x - ab2c     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

 Mar 7, 2018

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