If x, y ∈ R and n ∈ N, then the binomial theorem states that (x+y)^{n} = ^{n}C_{0}x^{n} + ^{n}C_{1 }x^{n1}y+ ^{n}C_{2} x^{n2 }y^{2} +…… … .. + ^{n}C_{r}x^{nr }y^{r} + ….. + ^{n}C_{n}y^{n }which can be written as Σ^{n}C_{r}x^{nr}y^{r}. This is also called as the binomial theorem formula which is used for solving many problems.
The (r +1)^{th}term in the expansion of expression (x+y)^{n} is called the general term and is given by T_{r+1 }= ^{n}C_{r}x^{nr}y^{r}
The term independent of x is obviously without x and is that value of r for which the exponent of x is zero.
The middle term of the binomial coefficient depends on thevalue of n. There can be two different cases according to whether n is even or n is odd.
The binomial coefficient of the middle term is the greatest binomial coefficient of the expansion.
Some of the standard binomial theorem formulas which should be memorized are listed below:
In order to compute numerically greatest term in a binomial expansion of (1+x)^{n}, find T_{r+1 }/ T_{r}= (n – r + 1)x/r. Then put the absolute value of x and find the value of r which is consistent with the inequality T_{r+1 }/ T_{r}> 1.
If the index n is other than a positive integer such as a negative integer or fraction, then the number of terms in the expansion of (1+x)^{n}is infinite.
The expansions in ascending powers of x are valid only if x is small. If x is large, i.e. x > 1 then it is convenient to expand in powers of 1/x which is then small.
The binomial expansion for the nth degree polynomial is given by:
Following expansions should be remembered for x < 1:
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