Question

How do you find the product of \[\left( x+1 \right)\left( {{x}^{2}}+x+1 \right)\]?

Answer 1

Hint: In this problem, we have to find the product of the given equations. To find the product, we can multiply the terms in the left parenthesis by each individual term right parenthesis. We can take the first term in the left parenthesis and multiply with the three terms in the right parenthesis, similarly we can take the second term in the left parenthesis and multiply with the three terms in the right parenthesis and finally add the terms to get a final answer.

Complete step by step answer:
We know that the given equation to be multiplied to get the product is,
 \[\left( x+1 \right)\left( {{x}^{2}}+x+1 \right)\]
We can multiply the terms in the left parenthesis by each individual term right parenthesis
We can take the first term in the left parenthesis and multiply with the three terms in the right parenthesis, similarly we can take the second term in the left parenthesis and multiply with the three terms in the right parenthesis.
\[\Rightarrow \left( x\times {{x}^{2}} \right)+\left( x\times x \right)+\left( x\times 1 \right)+\left( 1\times {{x}^{2}} \right)+\left( 1\times x \right)+\left( 1\times 1 \right)\]
Now we can multiply the above terms, we get
\[\Rightarrow {{x}^{3}}+{{x}^{2}}+x+{{x}^{2}}+x+1\]
Now we can add the terms in the above step, we get
\[\Rightarrow {{x}^{3}}+2{{x}^{2}}+2x+1\]

Therefore, the product of \[\left( x+1 \right)\left( {{x}^{2}}+x+1 \right)\] is \[{{x}^{3}}+2{{x}^{2}}+2x+1\].

Note: Students make mistakes while multiplying the individual terms, we can take the first term in the left parenthesis and multiply with the three terms in the right parenthesis, similarly we can take the second term in the left parenthesis and multiply with the three terms in the right parenthesis.