Before learning binomial expansion formulas, let us recall what is a "binomial". A binomial is an algebraic expression with two terms. For example, a + b, x - y, etc are binomials. We have a set of algebraic identities to find the expansion when a binomial is raised to exponents 2 and 3. For example, (a + b)2= a2+ 2ab + b2. But what if the exponents are bigger numbers? It is tedious to find the expansion manually. The binomial expansion formula eases this process. Let us learn the binomial expansion formula along with a few solved examples.
What AreBinomial Expansion Formulas?
As we discussed in the earlier section, the binomial expansion formulas are used to find the powers of the binomials which cannot be expanded using the algebraic identities. The binomial expansion formula involves binomial coefficients which are of the form\(\left(\begin{array}{l}n \\k\end{array}\right)\) (or) \(n_{ C_{k}}\)and it is calculated using the formula,\(\left(\begin{array}{l}n \\k\end{array}\right)\) =n! / [(n - k)! k!]. The binomial expansion formula is also known as the binomial theorem.Here are the binomial expansion formulas.
Binomial Expansion Formula of Natural Powers
Thisbinomial expansion formula gives the expansion of (x + y)nwhere 'n' is a natural number. The expansion of (x + y)nhas (n + 1) terms. This formula says:
(x+ y)n= nC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0yn
Here we usenC\(_k\) formula to calculate the binomial coefficients which saysnC\(_k\) = n! / [(n - k)! k!]. By applying this formula, the above binomial expansion formula can also be written as,
(x+ y)n= xn+ nxn - 1 y1+ [n(n - 1)/2!]xn-2 y2+ [n(n - 1)(n - 2)/3!]xn - 3y3+... + n x yn - 1+ yn
Note:If we observe just the coefficients, they are symmetric about the middle term. i.e., the first coefficient is the same as the last one, the second coefficient is as same as the one that is second from the last, etc.
Binomial Expansion Formula of Rational Powers
Thisbinomial expansion formula gives the expansion of (1+ x)nwhere 'n' is a rational number. This expansion has an infinite number of terms.
(1+ x)n= 1+ nx+ [n(n - 1)/2!]x2+ [n(n - 1)(n - 2)/3!]x3+...
Note:To apply this formula, the value of |x| should be less than 1.
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Examples UsingBinomial Expansion Formulas
Example 1:Find the expansion of(a + b)3.
Solution:
To find: (a + b)3
Using binomial expansion formula,
(x+ y)n= nC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0yn
(a + b)3= 3C\(_0\)a3+3C\(_1\)a(3 - 1)b +3C\(_2\)a(3 - 2) b2+3C\(_3\)a(3 - 3)b3
= ( 3! / [(3-0)!0!] )a3+( 3! / [(3-1)!1!] )a(3 - 1)b +( 3! / [(3-2)!2!] )a(3 - 2) b2+( 3! / [(3-3)!3!] )a(3 - 3)b3
= (1)a3+ (3) a2b + (3) a1b2+ (1)a0b3
= a3+ 3a2 b + 3ab2+ b3
Answer:(a + b)3= a3+ 3a2 b + 3ab2+ b3.
Example 2:Find the expansion of (x + y)6.
Solution:
Using the binomial expansion formula,
(x+ y)n= nC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0yn
(x + y)6=6C\(_0\) x6+6C\(_1\) x5y+6C\(_2\) x4y2+6C\(_3\) x3y3+6C\(_4\) x2y4+6C\(_5\) xy5+6C\(_6\) y6
=( 6! / [(6-0)!0!] )x6+ ( 6! / [(6-1)!1!] ) x5y+( 6! / [(6-2)!2!] ) x4y2+( 6! / [(6-3)!3!] ) x3y3+( 6! / [(6-4)!4!] ) x2y4+( 6! / [(6-5)!5!] ) xy5+( 6! / [(6-6)!6!] ) y6
= x6+ 6x5y + 15x4y2+ 20x3y3+ 15x2y4+ 6x y5+ y6
Answer:(x + y)6=x6+ 6x5y + 15x4y2+ 20x3y3+ 15x2y4+ 6x y5+ y6.
Example 3:Find the expansion of (3x + y)1/2upto the first three terms using the binomial expansion formula of rational exponents where\(\left|\dfrac y {3x}\right|\) < 1.
Solution:
(3x + y)1/2 = 3x (1 + y/(3x))1/2
Comparing (1 + y/(3x))1/2 with (1+ x)n, we have x =y/(3x) and n = 1/2.
The expansion of (1 + y/(3x))1/2 upto thefirst three termsusing the binomial expansion formula is,
1+ nx+ [n(n - 1)/2!]x2= 1 + (1/2) (y / (3x))+[(1/2) ((1/2)- 1)/2!](y / (3x))2
= 1 + y / (6x) - y2/ (72x2)
Thus, the expansion of 3x (1 + y/(3x))1/2 upto the first three terms is:
3x [1 + y / (6x) - y2/ (72x2) ] = 3x + y / 2 - y2/ (24x)
Answer:(3x + y)1/2 = 3x + y / 2 - y2/ (24x).
FAQs onBinomial Expansion Formulas
What Is Binomial Expansion In Maths?
Binomial expansion is to expand and write the terms which are equal to the natural number exponent of the sum or difference of two terms. For two terms x and y the binomial expansion to the power of n is(x+ y)n= nC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0yn. Here in this expansion the number of terms is equal to one more than the value of n.
What AreBinomial Expansion Formulas?
The binomial expansion formulas are used to find the expansion when a binomial is raised to a number. The binomial expansion formulas are:
- (x+ y)n= nC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0yn, where 'n' is a natural number andnC\(_k\) = n! / [(n - k)! k!].
(1+ x)n= 1+ nx+ [n(n - 1)/2!]x2+ [n(n - 1)(n - 2)/3!]x3+... , when 'n' is a rational number and here |x| < 1.
How To Derive Binomial Expansion Formula?
The binomial expansion formula is (x+ y)n= nC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0ynand it can be derived using mathematical induction. Here are the steps to do that.
- Step 1: Prove the formula for n = 1.
- Step 2: Assume that the formula is true for n = k.
- Step 3: Prove the formula for n = k.
What Are the Applications of theBinomial Expansion Formula?
The main use of the binomial expansion formula is to find the power of a binomial without actually multiplying the binominal by itself many times. This formula is used in many concepts of math such as algebra, calculus, combinatorics, etc.
How To Use theBinomial Expansion Formula?
The binomial expansion formula says the expansion of(x+ y)nisnC\(_0\)xny0+nC\(_1\)xn - 1 y1+nC\(_2\)xn-2 y2+nC\(_3\)xn - 3y3+ ... +nC\(_{n-1}\)x yn - 1+nC\(_n\)x0ynwherenC\(_k\) = n! / [(n - k)! k!]. If we have to find the expansion of (3a - 2b)7, we just substitute x = 3a, y = -2b and n = 7 in the above formula and simplify.
