SOLUTION: find c so that the polynomial will be the square of a bynomial cy2+6y+1 justify your answer

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Question 268873: find c so that the polynomial will be the square of a bynomial cy2+6y+1 justify your answer
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
find c so that the polynomial will be the square of a bynomial cy2+6y+1 justify your answer.

Assume A and B are such that

%28Ay%2BB%29%5E2=cy%5E2%2B6y%2B1

%28Ay%2BB%29%28Ay%2BB%29=cy%5E2%2B6y%2B1

A%5E2y%5E2%2B2ABy%2BB%5E2=cy%5E2%2B6y%2B1

To make that an identity we equate like terms on both sides:

system%28A%5E2y%5E2=cy%5E2%2C2ABy=6y%2CB%5E2=1%29

simplify

system%28A%5E2=c%2CAB=3%2CB%5E2=1%29

Solving the last equation 

B%5E2=1,

B%5E2-1=0

%28B-1%29%28B%2B1%29=0

Setting first factor = 0

B-1=0
B=1

Setting second factor = 0

B%2B1=0
B=-1

Substituting B=1 in the equation

AB=3

A%2A%281%29=3

A=3

Substituting that in

A%5E2=c

3%5E2=c

9=c

So

%28Ay%2BB%29%5E2=cy%5E2%2B6y%2B1

becomes

%283y%2B1%29%5E2=9y%5E2%2B6y%2B1

If you use B=-1, you end up with

%28-3y-1%29%5E2=9y%5E2%2B6y%2B1

Either way, C=9

Edwin