Multiple choice

The equation x4 - 5x3 + 3x2 + 35x - 70 = 0 has a root between

  1. 2 and 3

  2. 3 and 4

  3. 5 and 6

  4. None of these
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A Correct answer
Explanation

Let f(x) = x^4 - 5x^3 + 3x^2 + 35x - 70. Evaluating at x = 2 gives 16 - 40 + 12 + 70 - 70 = -12. Evaluating at x = 3 gives 81 - 135 + 27 + 105 - 70 = 8. Since the sign changes between 2 and 3, a root exists in that interval.

AI explanation

Using the intermediate value theorem, we evaluate the polynomial at the boundaries of the given intervals. Let f(x) = x^4 - 5x^3 + 3x^2 + 35x - 70. We find f(2) = 16 - 40 + 12 + 70 - 70 = -12 and f(3) = 81 - 135 + 27 + 105 - 70 = 8. Since f(2) is negative and f(3) is positive, a root lies between 2 and 3.

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