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The polynomial q(q + 1)(q + 2)(q + 3) + 1 is the square of what polynomial?

 Dec 15, 2019
 #1
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q(q + 1)(q + 2)(q + 3) + 1 = q^4 + 2q^3 + 3q^2 + q + 1 = (q^2 + 3q + 1)^2

 Dec 16, 2019
 #2
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+1

q(q + 1) (q + 2)(q + 3)  +  1   =

 

(q^2 + q)(q^2 + 5q + 6)  + 1   =

 

q^4  + 5q^3  + 6q^2

        +  q^3  + 5q^2 + 6q  + 1

_________________________

q^4  +  6q^3  + 11q^2 + 6q +  1

 

q^4  + 6q^3  + 9q^2 + 2q^2  + 6q +  1  =

 

(q^2 +3q)^2  + 2(q^2 + 3q)  +  1    =

 

([ q^2 + 3q ] + 1) ([ q^2  + 3q ]  + 1 )  =

 

(q^2  + 3q + 1)^2

 

 

cool cool cool

 Dec 16, 2019

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