Question

The sum of the coefficients in the expansion of $(1+2x-x^{2})^{20}$ is

• A. $2^{20}$
• B. $2^{21}$
• C. $2^{19}$
• D. $2^{40}$
• E. 2

Hint:

Algebra Tip: This "plug in 1" trick works for polynomials of ANY number of variables! For example, the sum of coefficients of $(3x - 2y + z)^5$ is just $(3 - 2 + 1)^5 = 2^5 = 32$.

Answer 1

The Correct Option is A

Concept:
The sum of all coefficients in the algebraic expansion of a polynomial $P(x)$ can be found simply by substituting $x = 1$ into the unexpanded expression. This collapses all variable terms, leaving only the arithmetic sum of the coefficients.

Step 1: Identify the polynomial expression.
The given polynomial to be expanded is: $$P(x) = (1 + 2x - x^2)^{20}$$

Step 2: Apply the rule for finding the sum of coefficients.
To find the sum of the coefficients, we set the variable $x$ to $1$: $$\text{Sum} = P(1)$$

Step 3: Substitute x = 1 into the expression.
Replace every instance of $x$ inside the parentheses with $1$: $$\text{Sum} = (1 + 2(1) - (1)^2)^{20}$$

Step 4: Simplify the expression inside the base.
Perform the basic arithmetic inside the parentheses: $$\text{Sum} = (1 + 2 - 1)^{20}$$ $$\text{Sum} = (3 - 1)^{20}$$ $$\text{Sum} = (2)^{20}$$

Step 5: State the final answer.
The fully evaluated expression gives the sum of all coefficients across the 41 terms of the expansion. $$\text{Sum} = 2^{20}$$ Hence the correct answer is (A) $2^{20$}.