Question

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

Answer 1

Hint: The given polynomial can be expanded as ${(1 + x - 3{x^2})^{2163}} = p(x) = \sum\limits_{n = 0}^{4326} {{a_n}{x^n}} $ where \[\{ {a_{0,}}{a_1},{a_2}, \cdots ,{a_{4325}},{a_{4326}}\} \] are coefficients of the following function. We have to put x such that the expansion is independent of $x$.

Complete step-by-step answer:
We are given ${(1 + x - 3{x^2})^{2163}}$ and we need to find the sum of all the coefficients in expansion. This is a case of multinomial expansion.
First we will show the general expansion of the given polynomial. This is shown as
${(1 + x - 3{x^2})^{2163}} = p(x) = \sum\limits_{n = 0}^{4326} {{a_n}{x^n}} $ … (1)
 $\sum\limits_{n = 0}^{4326} {{a_n}{x^n}} = {a_0}{x^0} + {a_1}{x^1} + {a_2}{x^2} + ................ + {a_{4326}}{x^{4326}}$ … (2)
Now, we need to find the value of \[{a_0} + {a_1} + {a_2} + \cdots {a_{4325}} + {a_{4326}}\]. This can be shown as follows
$S = \sum\limits_{n = 0}^{4326} {{a_n}} $
If we have to find the submission of the coefficients then we must take the value of $x$ such that the equation must become independent of $x$ and we get the equation corresponding to\[{a_0} + {a_1} + {a_2} + \cdots {a_{4325}} + {a_{4326}}\].
Thus, only for $x = 1$. Thus putting $x = 1$ in equation (2), we get,
$\sum\limits_{n = 0}^{4326} {{a_n}{{(1)}^n}} = {a_0}{1^0} + {a_1}{1^1} + {a_2}{1^2} + ................ + {a_{4326}}{1^{4326}}$
Now, putting $x = 1$in equation (1) we get,
${[1 + 1 - 3{(1)^2}]^{2163}} = \sum\limits_{n = 0}^{4326} {{a_n}{{(1)}^n}} = \sum\limits_{n = 0}^{4326} {{a_n}} $
Simplifying the above equation, we get,
$\sum\limits_{n = 0}^{4326} {{a_n} = } {( - 1)^{2163}}$
Since, we know -1 raises to power a negative number, we get -1. Therefore,
$\sum\limits_{n = 0}^{4326} {{a_n} = } {( - 1)^{2163}} = - 1$ ... (3)
Since, equation (2) and equation (3) are similar. We can say that,
\[{a_0} + {a_1} + {a_2} + \cdots {a_{4325}} + {a_{4326}} = - 1\]

Therefore, option (b) -1 is the correct option.

Note: Important points to remember while solving the questions on binomial expansion are:
1. The total number of terms in the expansion of \[{\left( {x + y} \right)^n}\;\] are (n+1).
2. The sum of exponents of x and y is always n.
While these properties are not true for multinomial expansion generally.