Activité 2
Question
1) Développer (a+b)2 et (a+b)3
2) Calculer : \(∑_{p=0}^2C_2^P a^{2-p} b^p\) et \(∑_{p=0}^3C_3^P a^{3-p} b^p\)
3) Que remarque-t-on ?
Solution
1) Développons :
\((a+b)^2= a^2+2ab+b^2\)
\((a+b)^3 = a^3+3a^2 b+3ab^2+b^3\)
2) Calculons :
\(∑_{p=0}^2C_2^P a^{2-p} b^p = C_2^0 a^2 b^0+C_2^1 a^1 b^1+ C_2^2 a^0 b^2\)
=\(1×a^2+2ab+1b^2\)
= \(a^2+2ab+b^2\)
\(∑_{p=0}^3 C_3^P a^{3-p} b^p= C_3^0 a^3 b^0+C_3^1 a^2 b^1+C_3^2 a^1 b^2+C_3^3 a^0 b^3\)
=\(1 a^3+3 a^2 b^1+3 a^1 b^2+1 b^3\)
=\(a^3+3 a^2 b+3ab^2+b^3\)
3) On remarque que :
\((a+b)^2=∑_{p=0}^2 C_2^P a^{2-p} b^p =a^2+2ab+b^2\)
\((a+b)^3=∑_{p=0}^3 C_3^P a^{3-p} b^p= a^3+3 a^2 b+3ab^2+ b^3\)