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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.
explanation: The document contains a math question along with its detailed solution. The question asks for the sum of the coefficients of the polynomial \((1 + x - 3x^2)^{1947}\), and the solution provides a step-by-step method to find the answer, which is \(-1\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Courses
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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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question along with its detailed solution. The question asks for the sum of the coefficients of the polynomial \((1 + x - 3x^2)^{1947}\), and the solution provides a step-by-step method to find the answer, which is \(-1\).
  
  Label: question-and-answer