Question

P(x)=x+x²+x³+.............. + x ²⁰²³. What number is P(-1) ?

• A. 2023
• B. -2023
• C. 0
• D. -1

Answer 1

Answer:

D. -1

The value of P(-1) is -1

• Substitute x = -1. Replace every x in the polynomial with -1 to get $P(-1) = (-1)^1 + (-1)^2 + (-1)^3 + \dots + (-1)^{2023}$

• Apply exponent rules. An odd power of -1 equals -1, and an even power of -1 equals +1

• Rewrite the series. The expression simplifies to $P(-1) = -1 + 1 - 1 + 1 - \dots - 1$

• Group into pairs. There are (2023) terms in total. The first (2022) terms form (1011) pairs of (-1 + 1), and each pair adds up to 0

• Find the remainder. After cancelling out the first (2022) terms, only the final odd term, $(-1)^{2023}$, remains. This gives the final answer of -1