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Find the coefficient of ${{\text{x}}^3}{\text{ in }}{\left( {1 + {\text{x + }}{{\text{x}}^2}} \right)^3}.$

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Last updated date: 24th Jul 2024
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Answer
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Hint: For finding the coefficient of ${{\text{x}}^3}$, we have to first expand the given algebraic expression. The given algebraic expression is in the form of ${\left( {{\text{a + b + c}}} \right)^3}$. So to get the term of different powers, we have to expand this expression. After expanding, collect the terms with power 3.

Complete step-by-step answer:
In the question, we have to find the coefficient of ${{\text{x}}^3}$. The algebraic expression given is:
${\left( {1 + {\text{x + }}{{\text{x}}^2}} \right)^3}.$
Now to get the coefficient of ${{\text{x}}^3}$, we have to first expand the given expression.
The given algebraic expression is in the form of ${\left( {{\text{a + b + c}}} \right)^3}$ and we know that:
${\left( {{\text{a + b + c}}} \right)^3} = {{\text{a}}^3} + {{\text{b}}^3} + {{\text{c}}^3} + 3\left( {{\text{a + b}}} \right)\left( {{\text{b + c}}} \right)\left( {{\text{c + a}}} \right).$
Putting the value of a, b and c in the above identity, we get:
${(1 + {\text{x + }}{{\text{x}}^2})^3} = {1^3} + {{\text{x}}^3} + {({{\text{x}}^2})^3} + 3\left( {{\text{1 + x}}} \right)\left( {{\text{x + }}{{\text{x}}^2}} \right)\left( {{{\text{x}}^2}{\text{ + 1}}} \right)$
On further expanding the above expression, we get:
\[
  {(1 + {\text{x + }}{{\text{x}}^2})^3} = {1^3} + {{\text{x}}^3} + {{\text{x}}^6} + 3\left( {{\text{1 + x}}} \right)\left( {{{\text{x}}^4} + {{\text{x}}^3} + {{\text{x}}^2} + {\text{x}}} \right) \\
  {(1 + {\text{x + }}{{\text{x}}^2})^3} = 1 + {{\text{x}}^3} + {{\text{x}}^6} + 3\left( {{{\text{x}}^5} + 2{{\text{x}}^4} + 2{{\text{x}}^3} + 2{{\text{x}}^2} + {\text{x}}} \right) \\
  {(1 + {\text{x + }}{{\text{x}}^2})^3} = 1 + {{\text{x}}^3} + {{\text{x}}^6} + 3{{\text{x}}^5} + 6{{\text{x}}^4} + 6{{\text{x}}^3} + 6{{\text{x}}^2} + 3{\text{x}} \\
  {(1 + {\text{x + }}{{\text{x}}^2})^3} = {{\text{x}}^6} + 3{{\text{x}}^5} + 6{{\text{x}}^4} + 7{{\text{x}}^3} + 6{{\text{x}}^2} + 3{\text{x + 1}}{\text{.}} \\
{\text{So, the final expression that we get is}}:
\{{(1 + {\text{x + }}{{\text{x}}^2})^3} = {{\text{x}}^6} + 3{{\text{x}}^5} + 6{{\text{x}}^4} + 7{{\text{x}}^3} + 6{{\text{x}}^2} + 3{\text{x + 1}}\]----- (1)
In the algebraic expression given by equation 1:
The coefficient of ${{\text{x}}^3}$ is 7.

Note: In this type of question where the polynomial is not given in expanded form. We have to first expand the given algebraic expression into standard polynomial form using required algebraic identities. After this, collect the terms having power 3. Its coefficient will be the required answer.