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The polynomial which results from the expansion of $(x^2+5x+6)^2+(px+q)(x^3+7x^2+3x)$ has degree $2$. Find $p+q$.

michaelcai  Sep 11, 2017
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(x^2+5x+6)^2+(px+q)(x^3+7x^2+3x)

 

x^4 + 10 x^3 + 37 x^2 + 60x + 36   +  px^4 + 7px^3 + 3px^2  + qx^3 + 7qx^2 + 3qx

 

( 1 + p)x^4  + (10 + 7p + q) x^3 + (37 + 3p + 7q)x^2 +( 60+ 3q)x + 36

 

If the resulting polynomial is of degree 2, then

 

1 + p   = 0    →  p  = -1

 

And

 

(10 + 7p + q)  = 0   

(10  - 7  + q)  = 0

3 + q  = 0

q  = -3

 

So

 

(x^2+5x+6)^2+(-1x-3)(x^3+7x^2+3x)  =  13 x^2 + 51 x + 36

 

So

 

p + q  =  ( -1 )  +  (- 3)   =  - 4

 

 

 

cool cool cool    

CPhill  Sep 11, 2017

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