Profile
Settings
Refer your friends
Sign out
The algebraic expansion of powers of a binomial is described by the binomial theorem or expansion. The polynomial “(a + b)n” can be expanded into a sum including terms of the form “axzyc,” where the exponents z and c are non-negative integers and z + c = n, and the coefficient of each term is a positive integer dependent on the values of n and b.
The Binomial theorem formula is extremely useful in determining the power of a binomial. The approach is simplified because it does not require the tedious process of multiplication.
Furthermore, applying the method aids in calculating the middle and general terms in a binomial expansion of a mathematical statement.
When the binomial (a +b)n is represented in expanded form using the binomial theorem, you’ll see the intermediate and general terms in this section.
The formula of binomial theorem in the form of general term:
It is known that there are (n + 1) terms in the binomial expansion pertaining to (a+b)n using the Binomial theorem formula.
Let us now state that T1, T2, T3, T4, T5… Tn+1 are the first, second, third, fourth, fifth,….. (n + 1)th terms in the (a+b)n expansion, in order.
Hence,
T1 = (nC0)aⁿ
T2 = (nC1)an-1.b
T3 = (nC2)an-2.b2
T4 = (nC3)an-3.b3
Tn+1 = (nCn)bn
In terms of generality, we have the following formula:
Tr+1 = (nCr)an-r. br
where 0 ≤ r ≤ n. Let’s have a look at an example.
Example:
Question: Find the 4th term in the binomial expansion of (4x–2y)⁶.
Answer: Given:
n = 6, a = 4x, b = – 2y. We’re having problems with the formula above.
Tr+1 = (nCr)an-r. br
T4 , r = 3 to find the fourth term. Thus,
T4 = T3+1 = (7/3) (4x)^6–3.(−2y)^3
= 6.5.4 / 1.2.3. (64x³). (-8y³)
= – 10,240.x³.y³
As a result, the fourth term in the expansion of (4x–2y)6 = – 10,240x3y3.
The formula for binomial theorem in the form of middle term:
If you want to expand the expression (a+b)n, remember that ‘n’ is an even integer, hence (n + 1) is an odd number. This means that terms pertaining to odd numbers will make up the binomial expansion.
The middle term of the binomial theorem formula will be equivalent to (n / 2 + 1)th term in this case.
Let’s say you extend (x+y)2, and the middle term becomes (2 / 2 + 1), which is equal to the second term. The (n2 + 1)th term can also be written as (n+2 / 2)th term.
When (a+b)n is the expression, you try to expand it, and because ‘n’ is an odd integer, (n + 1) will be an even number. As a result, there are two middle terms: (n+1 / 2)th term and (n+3 / 2)th term.
If you expand the expression (x+y)3, the middle terms are (3+1 / 2) = 2nd term and (3+3 / 2) = 3rd term, respectively.
Example:
Question: Find the middle terms in the expansion of (x / 4 + 2y)^11.
Solution: Given: n = 11 is an odd number, and we have two middle terms: (11+1 / 2) = 6th term and (11+3 / 2) = 7th term.
We are also experiencing,
a = x / 4, b = 2y, and n = 11 are the values.
As is well known,
Tr+1 = (n / r)an-r .br
To locate the sixth phrase,
T6 , r = 5. Thus,
T6 = T5+1 = (11 / 5)(x / 4)^11–5. (2y)5
= 11.10.9.8.7 / 1.2.3.4.5. (x⁶ / 4096) . (32y⁵)
= 231 / 64. x⁶. y⁵
Similarly,
T7 = T6+1 = (11 / 6)(x / 4)^11–6. (2y)⁶
= 11.10.9.8.7.6 / 1.2.3.4.5.6 .(x⁵ / 1024) . (729y⁶)
= 231 / 8 . x⁵. y⁶
Thus, the middle terms of the (x / 4 + 2y)11 expansion = 231 / 8.x5. y6.
Conclusion:
The Binomial Expansion is in its middle term. The expansion of (a + b)n comprises (n + 1) terms, as we know. The algebraic expansion of powers of a binomial is described by the binomial theorem or expansion. The Binomial theorem formula is extremely useful in determining the power of a binomial. The approach is simplified because it does not require the tedious process of multiplication.
Furthermore, applying the method aids in calculating the middle and general terms in a binomial expansion of a mathematical statement.
Let us now state that T1, T2, T3, T4, T5… Tn+1 are the first, second, third, fourth, fifth,….. (n + 1)th terms in the (a+b)n expansion, in order.
If you want to expand the expression (a+b)n, remember that ‘n’ is an even integer, hence (n + 1) is an odd number. This means that terms pertaining to odd numbers will make up the binomial expansion. The middle term of the binomial theorem formula will be equivalent to (n / 2 + 1)th term in this case.
