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Binomial Theorem

Binomial Expansion for Positive Integral Index

Do you know what is a Binomial Expansion? Well, as the name suggests, the binomial is an expression which has two terms and an operator like (+,-). An example of a binomial expression is a+b, 2x+4z etc. These binomial expressions can also have powers (degrees) and can be expanded to simplify the calculations. Let’s find out more in the section below.

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Binomial Expansion

Binomial theorem is a sorted way (or formula) of expanding expressions that are raised to some large power. Let us first understand what are binomial expressions with the help of some examples.

Example 1 (a+b)2

Suppose we have an expression (a+b)2Here (a+b)is an expression with an operator +, now we have to expand it. Well, we all know its formula

$${( a+b) }^{ 2 }={ a }^{ 2 } + { b }^{ 2 } + 2ab$$

This type of expansion of an expression is called binomial expansion.

Example 2 (a+b)3

The second expression is (a+b)3.  To expand this, we have the formula-

$$ {( a+b) }^{ 3 }={ a }^{ 3 }+{ b }^{ 3 }+3ab(a+b)$$

Example 3 (a+b)6

To expand this expression we have two ways. First, write (a+b) 6 times and multiply each one by one, like below:

$$(a+b)\times (a+b)\times (a+b)\times (a+b)\times (a+b)\times (a+b)$$

Another way is to divide 6 as 2*3. In that case first apply (a+b)2 formula first, then multiply that expression 3 times, i.e

$${ (a+b) }^{ 2 }={ a }^{ 2 }+{ b }^{ 2 }+2ab$$

And then multiply them three times. The above two methods are very lengthy, time consuming and chances of mistakes are also high. So to avoid such errors and mistakes we use binomial theorem.

Explanation of Binomial Theorem

Binomial Expansion

As stated earlier, the binomial theorem is used for the larger power of any expression. Let’s start with the formula-

\( (a+b)^{n}\)= \( {n} \choose {0} \)\( a^{n – 0} b^0\)+ \({n} \choose {1}\)\(a^{n – 1} b^1 + \) …. \( + {n} \choose {r} \)\(a^{n – r} b^r\)+ … +\({n} \choose {n-1}\)\(a^{1} b^{n-1}\)+\({n} \choose {n}\)\(a^{0} b^{n} \)

Above expression is an expansion of the binomial theorem. Some important points to remember of this theorem:

  • n must be positive integer
  • Binomial expansion always starts from 0 to the highest power of n. For example, if the value of n is 4 then expansion will start from 0 to 4.
  • C is called the combination. Here is its formula- \( {n} \choose {r} \) = \( \frac {n!}{n!(n-r)!} \)

Here n is always greater than r. For example- if n is 12 and r is 2,

\(n!=12!=12\times 11\times 10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\)
\(r!=2!=2\times 1\)
\((n-r)!=10!=10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\)
\(12!\div(10!\times 2!)=12\times 11\times 10!\div (10!\times 2)\)

On solving , the final answer is 66. This is the way to solve combinations and without combinations, it will be very difficult to solve binomial expansions.

  • Power of a is the subtraction of n and r. Power of b is always r. That’s why in the first expansion power of a is n-0 and b is 0. It can be seen in the expansion that the power of ‘b’ is increasing because the value of r is also increasing.

Conclusion

Binomial Theorem is always easier than multiplying the big and long expressions. To avoid error and mistakes, first practice the combinations formula with some examples, then try practising the binomial theorem. Binomial expansion is a simple combination of multiplication, division, addition and subtraction.

Solved Examples for You

Question 1: Sample Example of the Binomial Theorem.

\({ \left( 3x-2 \right) }^{ 10}={ _{ 10 }{ C }_{ 0 }{ \left( 3x \right) }^{ 10-0 } }{ \left( -2 \right) }^{ 0 }+{ _{ 10 }{ C }_{ 1 }{ \left( 3x \right) }^{ 10-1 } }{ \left( -2 \right) }^{ 1 }+{ _{ 10 }{ C }_{ 2 }{ \left( 3x \right) }^{ 10-2 } }{ \left( -2 \right) }^{ 2 }\)

\(\qquad  \quad \quad \quad \quad \quad \quad \quad \quad +{ _{ 10 }{ C }_{ 3 }{ \left( 3x \right) }^{ 10-3 } }{ \left( -2 \right) }^{ 3 }+{ _{ 10 }{ C }_{ 4 }{ \left( 3x \right) }^{ 10-4 } }{ \left( -2 \right) }^{ 4 }+{ _{ 10 }{ C }_{ 5 }{ \left( 3x \right) }^{ 10-5 } }{ \left( -2 \right) }^{ 5 }\)

\( \qquad \qquad \qquad \qquad +{ _{ 10 }{ C }_{ 6 }{ \left( 3x \right) }^{ 10-6 } }{ \left( -2 \right) }^{ 6 }+{ _{ 10 }{ C }_{ 7 }{ \left( 3x \right) }^{ 10-7 } }{ \left( -2 \right) }^{ 7 }+{ _{ 10 }{ C }_{ 8 }{ \left( 3x \right) }^{ 10-8 } }{ \left( -2 \right) }^{ 8 }\)

\(\qquad \qquad \qquad \qquad \quad +{ _{ 10 }{ C }_{ 9 }{ \left( 3x \right) }^{ 10-9 } }{ \left( -2 \right) }^{ 9 }+{ _{ 10 }{ C }_{ 10 }{ \left( 3x \right) }^{ 10-10 } }{ \left( -2 \right) }^{ 10 }\)

Answer : In this example we are expanding (3x – 2)10. According to the formula, here a is 3x and b is -2 and n is 10. So we are expanding it from 0 to 10. The value of n varies from 0 to 10. On solving the above combinations and values, here is the consolidated data:

$${ \left( 3x-2 \right) }^{ 10 }=(1)(59049){ x }^{ 10 }(1)+(10)(19683){ x }^{ 9 }(–2)+(45)(6561){ x }^{ 8 }(4)\\ \quad \quad \quad \quad \quad \quad \quad \quad +(120)(2187){ x }^{ 7 }(–8)+(210)(729){ x }^{ 6 }(16)+(252)(243){ x }^{ 5 }(–32)\\ \quad \quad \quad \quad \quad \quad \quad \quad +(210)(81){ x }^{ 4 }(64)+(120)(27){ x }^{ 3 }(–128)+(45)(9){ x }^{ 2 }(256)\\ \quad \quad \quad \quad \quad \quad \quad \quad +(10)(3)x(–512)+(1)(1)(1)(1024)$$

And here is the final result,

$${ \left( 3x-2 \right) }^{ 10 }=59049{ x }^{ 10 }–393660{ x }^{ 9 }+1180980{ x }^{ 8 }–2099520{ x }^{ 7 }\\ \qquad \qquad \quad +2449440{ x }^{ 6 }–1959552{ x }^{ 5 }+1088640{ x }^{ 4 }–414720{ x }^{ 3 }\\ \qquad \qquad \quad +103680{ x }^{ 2 }–15360x+1024$$

Question 2: What is meant by expanding a polynomial?

Answer: One can do expansion of a polynomial expression can by repeatedly replacing subexpressions that facilitate the multiplication of two other subexpressions, of which at least one is an addition, due to the equivalent sum of products. Moreover, this continues until the expression changes into a sum of (repeated) products.

Question 3: What is meant by binomial math theorem?

Answer: Binomial theorem means that for any positive integer n, the expression of the nth power of the sum of two numbers a and b may take place as the sum of n + 1 terms of the particular form.

Question 4: What is meant by n and r in a binomial theorem?

Answer: In n – r + 1, n happens to be the exponent of the binomial while r represents the term number.

Question 5: Give an example of a binomial?

Answer: Binomial refers to a polynomial equation with two terms whose joining usually takes place by a plus or minus sign. The use of binomials takes place in algebra. For example, 3x + 4 happens to be a binomial as well as a polynomial.  2a (a+b) 2 is another good example of a binomial.

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