Respuesta :
We are given the expression below to expand:
[tex](5x-2)^6[/tex]To find the coefficient of the expansion we will need to use the formula below. The formula below indicates the binomial expansion formula
[tex]\sum ^n_{k\mathop=0}C^n_ka^kb^{n-k}[/tex]This will enable us to find the particular term we want, then we can pick out the coefficient of the term.
The parameters are as follows
n=6,k=0,1,2,3,4,5,6, a=5x i.e the first term and b =-2 i.e the second term
We will then apply parameters to the formula above.
[tex]^6C_0(5x)^6(-2)^0+^6C_1(5x)^5(-2)^1+^6C_2(5x)^4(-2)^2+^6C_3(5x)^3(-2)^3+^6C_4(5x)^2(-2)^4+^6C_5(5x)^1(-2)^5_{_{}}+^6C_6(5x)^0(-2)^6[/tex]The expression above gives the interpretation of the binomial formula when the parameters are inserted.
Since we are looking for the coefficient of the x^2 we narrow it down to the term below, gotten from the expression above
[tex]^6C_4(5x)^2(-2)^4[/tex]We then simplify the above to get the required answer
[tex]\begin{gathered} ^6C_4(5x)^2(-2)^4 \\ =\frac{6!}{(6-4)!4!}\times25x^2\times16 \\ =\frac{6!}{2!4!}\times400x^2 \\ =\frac{6\times5\times4!}{2\times1\times4!}\times400x^2 \\ =3\times5\times400x^2 \\ =6000x^2 \end{gathered}[/tex]Therefore, the coefficient of the x^2 term is
[tex]6000[/tex]