Question

CMR:

      (x+y)(x+2y)(x+3y)(x+4y)+\(^{y^4}\)Là số chính phương

Answer 1

\(\left(x+y\right)\left(x+2y\right)\left(x+3y\right)\left(x+4y\right)+y^4\)

\(=\text{[}\left(x+y\right)\left(x+4y\right)\text{]}\text{[}\left(x+2y\right)\left(x+3y\right)\text{]}+y^4\)

\(=\left(x^2+5xy+4y^2\right)\left(x^2+5xy+6y^2\right)+y^4\)

\(=\text{[}\left(x^2+5xy\right)+4y^2\text{]}\text{[}\left(x^2+5xy\right)+6y^2\text{]}+y^4\)

\(=\left(x^2+5xy\right)^2+10y^2\left(x^2+5xy\right)+25y^4=\left(x^2+5xy+5y^2\right)\)

Vậy đề bài là số chính phương.