Answer: [tex]a^4+8a^3+24a^2+32a+16[/tex]
Step-by-step explanation:
The binomial expansion of [tex](a+b)^n=^nC_0a^nb^{0}+^nC_1a^{n-1}b^1+^nC_2a^{n-2}b^2+.........+^nC_na^0b^n[/tex]
For [tex](a+2)^4[/tex] , n=4 and b= 2 , we have
Then, the binomial expansion of [tex](a+2)^4[/tex] will be :
[tex](a+2)^4=^4C_0a^4(2)^{0}+^4C_1a^{4-1}2^1+^4C_2a^{4-2}2^2++^4C_3a^{4-3}2^3++^4C_4a^{0}2^4\\\\=(1)a^4+(4)a^3(2)+\dfrac{4!}{2!(4-2)!}a^2(4)+(4)a(8)+(1)(16)[/tex]
∵ [tex]^nC_0=^nC_n=0[/tex] and [tex]^nC_1=^nC_{n-1}=n[/tex]
[tex]=a^4+8a^3+(6)a^2(4)+32a+16\\\\=a^4+8a^3+24a^2+32a+16[/tex]
Hence, the binomial expansion of [tex](a+2)^4=a^4+8a^3+24a^2+32a+16[/tex]
