Multiple choice

Find the number of rational roots of the equation $2x^{99}+3x^{98}+2x^{97}+...+2x+3=0$.

  1. $2$
  2. $1$
  3. $3$
  4. $0$
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B Correct answer
Explanation

By the Rational Root Theorem, any rational root p/q must have p as a factor of the constant term (3) and q as a factor of the leading coefficient (2). Testing values like -1, -3/2, etc., shows only -1 is a root.

AI explanation

Grouping the terms of the equation 2x^99 + 3x^98 + 2x^97 + ... + 2x + 3 = 0 into pairs from highest to lowest degree gives (2x^99 + 3x^98) + (2x^97 + 3x^96) + ... + (2x + 3) = 0. Factoring out 2x + 3 from the entire expression leaves (2x + 3)(x^98 + x^96 + ... + 1) = 0. The only rational root comes from setting the linear factor to zero, which gives x = -3/2. The other factor evaluated at x = -1 or x = 1 does not equal zero, confirming there is exactly 1 rational root.

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