Answer
Verified
478.5k+ views
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.
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.
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
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE