Question

How do you simplify \[\left( x+1 \right)\left( x-1 \right)\] ?

Answer 1

Hint: In the given question, we have been asked to simplify the equation. By simplifying the equation, we refer to multiplication of the terms within the bracket with each other that mean each term of one bracket should be multiplied by each term of the other bracket in order to simplify the equation. And then we have to combine the like terms by addition or subtraction and then apply the mathematical operations such as addition, subtraction, multiplication and division in the equation to get the simplified version of the given question. This way we can solve the given question.

Formula used:
The distributive property of multiplication:
\[\Rightarrow \]\[\left( a+b \right)\left( c-d \right)=a\times \left( c-d \right)+b\left( c-d \right)\]
Property of difference of square pattern:
\[\Rightarrow \left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]

Complete step by step answer:
We have the given expression:
\[\left( x+1 \right)\left( x-1 \right)\]
Applying the distributive property i.e. \[\left( a+b \right)\left( c-d \right)=a\times \left( c-d \right)+b\left( c-d \right)\]
\[\left( x+1 \right)\left( x-1 \right)=x\times \left( x-1 \right)+1\times \left( x-1 \right)\]
Applying the distributive property again in the right-hand side of the expression, we get
\[\left( x+1 \right)\left( x-1 \right)=\left[ \left( x\times x \right)-\left( x\times 1 \right) \right]+\left[ \left( 1\times x \right)-\left( 1\times 1 \right) \right]\]
Simplifying the terms in the right-hand side, we obtain
\[\left( x+1 \right)\left( x-1 \right)={{x}^{2}}-x+x-1\]
Combining the like terms, we obtain
\[\left( x+1 \right)\left( x-1 \right)={{x}^{2}}-1\]

Therefore, \[{{x}^{2}}-1\] is the required simplification of the given expression i.e. \[\left( x+1 \right)\left( x-1 \right)\].

Note:The expression given in the question is the binomial expression. As we saw the solution, it fits the difference of the square pattern i.e. \[\left( x+1 \right)\left( x-1 \right)={{x}^{2}}-{{1}^{2}}={{x}^{2}}-1\]. So whenever we see a product binomial of the form, \[\left( a+b \right)\left( a-b \right)\], we can immediately put the property of difference of square pattern and therefore the expansion or simplification of \[\left( a+b \right)\left( a-b \right)\] is \[{{a}^{2}}-{{b}^{2}}\].it is always important to carefully examining all the terms while expanding the terms, we should not do calculation mistakes and don’t neglect any of the term as it will give wrong answer.