SOLUTION: Please help me solve this problem, - Q- {For what values of p and q , the expession X^4 - 14x^3 + 71x^2 + px + q is a perfect square? } I have problem in this type of quest

Algebra ->  Square-cubic-other-roots -> SOLUTION: Please help me solve this problem, - Q- {For what values of p and q , the expession X^4 - 14x^3 + 71x^2 + px + q is a perfect square? } I have problem in this type of quest      Log On


   

Question 726930: Please help me solve this problem,
- Q- {For what values of p and q , the expession X^4 - 14x^3 + 71x^2 + px + q is a perfect square? }
I have problem in this type of questions.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There must be some quadratic polynomial ax%5E2%2Bbx%2Bc that squares to yield
x%5E4+-+14x%5E3+%2B+71x%5E2+%2B+px+%2B+q+ .
Since the leading coefficient in x%5E4+-+14x%5E3+%2B+71x%5E2+%2B+px+%2B+q+ is 1 ,
we know that a=1 , but we still have to find b and c.
By painfully multiplying we find that

If the polynomials x%5E4+-+14x%5E3+%2B+71x%5E2+%2B+px+%2B+q+ and x%5E4+%2B2bx%5E3+%2B%28b%5E2%2B2c%29x%5E2+%2B2bcx+%2B+c%5E2
are the same, then the coefficients are the same
2b=-14
b%5E2%2B2c=71
p=2bc and
q=c%5E2
The first two equations let us find b and c .
We can use the last two to calculate p and q from b and c.
2b=-14 --> b=-7
b%5E2%2B2c=71 --> %28-7%29%5E2%2B2c=71 --> 49%2B2c=71 --> 2c=22 --> c=11
p=2bc --> p=2%28-7%29%2811%29 --> highlight%28p=-154%29
q=c%5E2 --> q=11%5E2 --> highlight%28q=121%29