Answer

Verified

406.5k+ views

**Hint:**Binomial theorem is a method used to expand a binomial term that is raised to some power of positive integer. According to binomial theorem, the nth power of the sum of two numbers (say a and b) can be expressed (expanded) as the sum or series of (n+1) terms, provided that ‘n’ is a positive integer.

**Formula used:**

${{(x+y)}^{n}}=\sum\limits_{i=0}^{n}{{}^{n}{{C}_{i}}{{x}^{n-i}}{{y}^{i}}}$,

where x and y are real numbers and n is a positive integer (a natural number).

${}^{n}{{C}_{i}}=\dfrac{n!}{i!(n-i)!}$

**Complete step by step answer:**

Let us first understand what is the binomial theorem.Binomial theorem is a method used to expand a binomial term that is raised to some power of positive integer. According to binomial theorem, the nth power of the sum of two numbers (say a and b) can be expressed (expanded) as the sum or series of (n+1) terms, provided that ‘n’ is a positive integer.

Suppose we have an expression ${{(x+y)}^{n}}$, where x and y are real numbers and n is a positive integer (a natural number).

Then, the binomial expansion of the above expression is given as

${{(x+y)}^{n}}=\sum\limits_{i=0}^{n}{{}^{n}{{C}_{i}}{{x}^{n-i}}{{y}^{i}}}$

Here, i is a natural number taking values from 0 to n.

When we expand the summation we get that ${{(x+y)}^{n}}={}^{n}{{C}_{0}}{{x}^{n-0}}{{y}^{0}}+{}^{n}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}+{}^{n}{{C}_{2}}{{x}^{n-2}}{{y}^{2}}+.......+{}^{n}{{C}_{n-1}}{{x}^{n-(n-1)}}{{y}^{n-1}}+{}^{n}{{C}_{n}}{{x}^{n-n}}{{y}^{n}}$.

In the given question, $n=4$,

Therefore, the given expression can expanded, with the help of binomial theorem as

${{(2x-1)}^{4}}={}^{4}{{C}_{0}}{{(2x)}^{4-0}}{{(-1)}^{0}}+{}^{4}{{C}_{1}}{{(2x)}^{4-1}}{{(-1)}^{1}}+{}^{4}{{C}_{2}}{{(2x)}^{4-2}}{{(-1)}^{2}}+{}^{4}{{C}_{3}}{{(2x)}^{4-3}}{{(-1)}^{3}}+{}^{4}{{C}_{4}}{{(2x)}^{4-4}}{{(-1)}^{4}}$

This equation can be further simplified to

${{(2x-1)}^{4}}={}^{4}{{C}_{0}}{{(2x)}^{4}}{{(-1)}^{0}}+{}^{4}{{C}_{1}}{{(2x)}^{3}}{{(-1)}^{1}}+{}^{4}{{C}_{2}}{{(2x)}^{2}}{{(-1)}^{2}}+{}^{4}{{C}_{3}}{{(2x)}^{1}}{{(-1)}^{3}}+{}^{4}{{C}_{4}}{{(2x)}^{0}}{{(-1)}^{4}}$

$\Rightarrow {{(2x-1)}^{4}}={}^{4}{{C}_{0}}(16{{x}^{4}})-{}^{4}{{C}_{1}}(8{{x}^{3}})+{}^{4}{{C}_{2}}(4{{x}^{2}})-{}^{4}{{C}_{3}}(2x)+{}^{4}{{C}_{4}}(1)$ ….. (i)

Now, we shall use the formula ${}^{n}{{C}_{i}}=\dfrac{n!}{i!(n-i)!}$

Therefore, equation (i) can be simplified to

${{(2x-1)}^{4}}=\dfrac{4!}{0!(4-0)!}(16{{x}^{4}})-\dfrac{4!}{1!(4-1)!}(8{{x}^{3}})+\dfrac{4!}{2!(4-2)!}(4{{x}^{2}})-\dfrac{4!}{3!(4-3)!}(2x)+\dfrac{4!}{4!(4-4)!}(1)$

With this, we get that

${{(2x-1)}^{4}}=(1)(16{{x}^{4}})-\dfrac{4!}{1!3!}(8{{x}^{3}})+\dfrac{4!}{2!2!}(4{{x}^{2}})-\dfrac{4!}{3!1!}(2x)+\dfrac{4!}{4!0!}(1)$

$\Rightarrow {{(2x-1)}^{4}}=16{{x}^{4}}-(4)8{{x}^{3}}+\left( \dfrac{4\times 3}{2} \right)(4{{x}^{2}})-(4)(2x)+(1)$

Finally,

$\therefore {{(2x-1)}^{4}}=16{{x}^{4}}-32{{x}^{3}}+24{{x}^{2}}-8x+1$

Hence, we found the expansion of the given expression with the help of binomial theorem.

**Note:**when we expand an expression with the help of binomial theorem, the series consists of (n+1) terms. If you do not use the formula of combination ${}^{n}{{C}_{i}}$, then you can make use of Pascal's triangle and select the row that has (n+1) elements (numbers).

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

At which age domestication of animals started A Neolithic class 11 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE