Multiple choice

The number of real roots of equation x8 – x5 + x2 – x + 2 = 0 is

  1. 2

  2. 4

  3. 6

  4. 0
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D Correct answer
Explanation

Let f(x) = x^8 - x^5 + x^2 - x + 2. For x >= 1, x^5(x^3 - 1) + x(x - 1) + 2 > 0. For 0 <= x < 1, x^8 + x^2(1 - x^3) + (2 - x) > 0. For x < 0, let y = -x, then y^8 + y^5 + y^2 + y + 2 > 0. Since the function is always positive, there are no real roots.

AI explanation

Divide the polynomial x^8 - x^5 + x^2 - x + 2 by x^2 to carefully analyze its signs, noting that for x < 0, all terms are positive, meaning there are no negative roots. For x = 0, the equation evaluates to 2, and for x = 1, it evaluates to 2. For x > 1, the x^8 term dominates the -x^5 term, and the x^2 term dominates the -x term, ensuring the expression is always strictly positive. Since the polynomial is always greater than zero for all real values of x, it has 0 real roots.

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