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# The sum of coefficients of the polynomial ${\left( {1 + x - 3{x^2}} \right)^{1947}}$ isA. 0B. 1C. $- 1$D. None of these

Last updated date: 20th Jun 2024
Total views: 373.8k
Views today: 4.73k
Verified
373.8k+ views
Hint: Here we need to find the sum of the coefficients in the given expansion. For that, we will expand the given expression and assume all the coefficients to be variables. Then we will substitute the value of the given variable to be one on both sides of the equation. From there, we will get the value of the sum of coefficients in the given polynomial expansion.

Complete step by step solution:
The given expression is ${\left( {1 + x - 3{x^2}} \right)^{1947}}$.
Now, we will expand the given expression. For that, we will assume the coefficients in the polynomial expansion to be ${a_0}$, ${a_1}$, ${a_2}$, ${a_3}$, …….
Therefore, we can write the expansion as
${\left( {1 + x - 3{x^2}} \right)^{1947}} = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ......$
Now, we will put the value of $x$ as 1 on both sides of this equation. Therefore, we get
$\Rightarrow {\left( {1 + 1 - 3 \cdot {1^2}} \right)^{1947}} = {a_0} + {a_1} \cdot 1 + {a_2} \cdot {1^2} + {a_3} \cdot {1^3} + ......$
On adding and subtracting the terms inside the bracket, we get
$\Rightarrow {\left( { - 1} \right)^{1947}} = {a_0} + {a_1} + {a_2} + {a_3} + ......$
On further simplification, we get
$\Rightarrow - 1 = {a_0} + {a_1} + {a_2} + {a_3} + ......$

Hence, the sum of the coefficients in the given polynomial expansion is equal to $- 1$.
Hence, the correct option is option C.

Note:
A polynomial is defined as an expression, which consists of variables, exponents, and constants that are combined together using the mathematical operations like subtraction, addition, multiplication and division. A polynomial is expanded if no variable appears within parentheses and all like terms have been simplified or combined. We need to keep in mind to expand a polynomial, we multiply its factors (often by using the distributive property) or we perform the indicated operations and then we combine all the like terms to get the final expansion.
The text contains a mathematical question about finding the sum of the coefficients of a polynomial, and it provides a detailed solution leading to the answer.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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# The sum of coefficients of the polynomial ${\left( {1 + x - 3{x^2}} \right)^{1947}}$ isA. 0B. 1C. $- 1$D. None of these

Last updated date: 20th Jun 2024
Total views: 373.8k
Views today: 4.73k
Verified
373.8k+ views
Hint: Here we need to find the sum of the coefficients in the given expansion. For that, we will expand the given expression and assume all the coefficients to be variables. Then we will substitute the value of the given variable to be one on both sides of the equation. From there, we will get the value of the sum of coefficients in the given polynomial expansion.

Complete step by step solution:
The given expression is ${\left( {1 + x - 3{x^2}} \right)^{1947}}$.
Now, we will expand the given expression. For that, we will assume the coefficients in the polynomial expansion to be ${a_0}$, ${a_1}$, ${a_2}$, ${a_3}$, …….
Therefore, we can write the expansion as
${\left( {1 + x - 3{x^2}} \right)^{1947}} = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ......$
Now, we will put the value of $x$ as 1 on both sides of this equation. Therefore, we get
$\Rightarrow {\left( {1 + 1 - 3 \cdot {1^2}} \right)^{1947}} = {a_0} + {a_1} \cdot 1 + {a_2} \cdot {1^2} + {a_3} \cdot {1^3} + ......$
On adding and subtracting the terms inside the bracket, we get
$\Rightarrow {\left( { - 1} \right)^{1947}} = {a_0} + {a_1} + {a_2} + {a_3} + ......$
On further simplification, we get
$\Rightarrow - 1 = {a_0} + {a_1} + {a_2} + {a_3} + ......$

Hence, the sum of the coefficients in the given polynomial expansion is equal to $- 1$.
Hence, the correct option is option C.

Note:
A polynomial is defined as an expression, which consists of variables, exponents, and constants that are combined together using the mathematical operations like subtraction, addition, multiplication and division. A polynomial is expanded if no variable appears within parentheses and all like terms have been simplified or combined. We need to keep in mind to expand a polynomial, we multiply its factors (often by using the distributive property) or we perform the indicated operations and then we combine all the like terms to get the final expansion.
user
The text contains a mathematical question about finding the sum of the coefficients of a polynomial, and it provides a detailed solution leading to the answer.

TYPE: SINGLE_QUESTION_WITH_ANSWER