Czynniki

[pawel435]

Rozłóż na czynniki wielomiany:
\(\displaystyle{ x ^{4} +3x ^{3} -15x ^{2} -19x+30}\)
\(\displaystyle{ x ^{3} (x ^{2} -7) ^{2} -36x}\)
\(\displaystyle{ x ^{6} -1}\)
\(\displaystyle{ x ^{8} +x ^{4} +1}\)
\(\displaystyle{ x ^{4} -3x ^{2}+9}\)

[adek05]

Ukryta treść:    

1)
\(\displaystyle{ = (x-1)(x^{3}+4x^{2}-11x-30) = (x-1)(x-3)(x^{2}+7x+10) = (x-1)(x-3)(x+2)(x+5)}\)
2)
\(\displaystyle{ = x(x^{2}(x^{2}-7)^{2} -36) = x(x^{3} - 7x+6)(x^{3} -7x -6) = x(x-1)(x^{2}+x-6)(x+1)(x^{2}-x-6)}\)
3)
\(\displaystyle{ =(x^{2}-1)(x^4+x^2+1) = (x+1)(x-1)[(x^{2} +1)^{2} - x^{2}] = (x+1)(x-1)(x^{2} -x +1)(x^{2}+x+1)}\)
4)
\(\displaystyle{ = (x^{4}+1)^{2} - x^{4} = (x^{4}-x^{2} + 1)(x^{4} + x^{2} + 1) =}\)
I dalej podobnie w nawiasach dojść do różnicy kwadratów.
5)
\(\displaystyle{ = (x^2 +3)^{2} - 9x^{2}}\)
I z różnicy kwadratów.