Question

The sum of coefficients in the polynomial expansion of ${\left( {1 + x - 3{x^2}} \right)^{2163}}$ is
A.1
B.-1
C.0
D.None of these

Answer 1

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 and then we will substitute the value of the given variable to be one in 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 answer:
The given expression is ${\left( {1 + x - 3{x^2}} \right)^{2163}}$.
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
$ \Rightarrow {\left( {1 + x - 3{x^2}} \right)^{2163}} = {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.
$ \Rightarrow {\left( {1 + 1 - 3 \cdot {1^2}} \right)^{2163}} = {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)^{2163}} = {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 B.

Note: Since we have expanded the given expression or given polynomial. A polynomial is said to be expanded if no variable appears within parentheses and all like terms have been simplified or combined.
Remember 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. 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.