Question

Expand \[\left( x – 1 \right)^{4}\]

Answer 1

Hint: In this question, we need to expand the given algebraic expression \[\left( x – 1 \right)^{4}\]. We can expand the given expression by using the binomial expansion method. Binomial expression is nothing but the expression contains two or more terms connected by mathematical operations like addition, subtraction etc… By using the binomial expansion formula, we can expand the given expression easily.
Formula used :
Binomial formula to expand \[\left( a – b \right){{}^{n}}\] is \[{{}{{}^{n}}}C_{0} a{{}^{n}}b^{0} - {{}{{}^{n}}}C_{1}a^{n – 1}\ b^{1} + {{}^{n}}C_{2} a^{n – 2}\ b^{2}-{{}^{n}}C_{3} a^{n – 3}\ b^{3}\ + \ldots\]

Complete step-by-step solution:
Given,
\[\left( x – 1 \right)^{4}\]
We can expand the given expression in the binomial expansion method.
\[\left( a – b \right){{}^{n}} ={{}^{n}}C_{0} a{{}^{n}}b^{0} -{{}^{n}}C_{1} a^{(n – 1)}b^{1} + {{}^{n}}C_{2} a^{n – 2}b^{2}\ -{{}^{n}}C_{3} a^{n – 3}b^{3} + \ldots\]
Here \[a = x\] and \[b = 1\]
By applying the formula,
We get,
\[\left( x – 1 \right)^{4} = {{}^{4}}C_{0} x^{4}\left( 1 \right)^{0} - {{}^{4}}C_{1} x^{4 – 1}\left( 1 \right)^{1} +{{}^{4}}C_{2} x^{4 – 2}\left( 1 \right)^{2}\ -{{}^{4}}C_{3} x^{4 – 3}\left( 1 \right)^{3} +{{}^{4}}C_{4} x^{4 – 4}\left( 1 \right)^{4}\]
We know that \[{{}^{4}}C_{0}= {{}^{4}}C_{4}= 1\], \[{{}^{4}}C_{1}={{}^{4}}C_{3} = 4\] and also \[{{}^{4}}C_{2} = 6\]
On Simplifying,
We get,
\[\left( x – 1 \right)^{4} = 1\left( x^{4} \right)\left( 1 \right) – 4\left( x^{3} \right)\left( 1 \right) + 6\left( x^{2} \right)\left( 1 \right) – 4\left( x^{1} \right)\left( 1 \right) + 1\left( x^{0} \right)\left( 1 \right)\]
We know that \[x^{0}\] is \[1\]
\[= x^{4} – 4x^{3} + 6x^{2} – 4x + 1\]
Thus we get
\[\left( x – 1 \right)^{4} = x^{4} – 4x^{3} + 6x^{2} – 4x + 1\]
Final answer :
The expansion of \[\left( x – 1 \right)^{4}\ \] is \[x^{4} – 4x^{3} + 6x^{2} – 4x + 1\]

Note: An algebraic expression is nothing up it is built up with integers, constants, variables and mathematical operations (addition, subtraction, multiplication, division etc… ) In mathematics, a symbol (letter) which doesn’t have a value is called a variable. Similarly which has a fixed value is called constant. Expanding algebraic expression is combining one or more variables or numbers by performing the given algebraic operations. Binomial theorem is a powerful tool used in expanding in the concepts of algebra, probability etc..