Here we can find the coefficient of x4 in the given expression.
We know that
Binomial theorem for Positive Integral index
If a, b, are any two real numbers and n any natural number then
(a+b)n =
C0anb° +
C1an−1b +
C2an−2b2 + ... +
Cran−rbr + ... +
Cnx0bn ,
where
Cr =
Here
C0,C1,C2 ...
Cn are called binomial coefficients
i.e.,
(a+b)n =
Cran−rbr Given
(1+x+x2)3 Considering the expansion of
(1+x+x2)3 in the form
((1+x)+x2)3 We have
(1+x)3+3(1+x)2x2+3(1+x)x4+x6.
(Since, by definition of binomial theorem)
Here
x4 comes in the second and third term only.
Adding its coefficients, 3 + 3 = 6.
Therefore,
The coefficient of x4 in the expansion of
(1+x+x2)3 is 6.