Zadanie-trojkata pascala

[IchBinHier]

W dzialaniu \(\displaystyle{ (2+px)^{5}}\) wspolczynnik przy \(\displaystyle{ x^{3}}\) to 135. Oblicz wartoscl dla p a nastepnie wartosc wspolczynnika przy \(\displaystyle{ x^{4}}\)

[natkoza]

\(\displaystyle{ (2+px)^5=2^5+5\cdot 2^4px+10\cdot 2^3(px)^2+10\cdot 2^2(px)^3+5\cdot 2(px)^4+(px)^5=32+80px+80p^2x^2+40p^3x^3+10p^4x^4+p^5x^5\\
40p^3x^3=135\\
p^3=3,375\\
p=\frac{3}{2}}\)

[wb]

\(\displaystyle{ {5 \choose 2} 2^2 (px)^3=135x^3 \\ \\ {5 \choose 2} 4 p^3=135 \\ p^3= \frac{27}{8} \\ p= \frac{3}{2}}\)




\(\displaystyle{ {5 \choose 4} 2^1 ( \frac{3}{2}x )^4=5 2 \frac{81}{16}x^4= \frac{405}{8}x^4}\)