Multiple choice

The number of roots common between the two equations x3 + 3x2 + 4x + 5 = 0 and x3 + 2x2 + 7x + 3 = 0 is

  1. 0

  2. 1

  3. 2

  4. 3
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A Correct answer
Explanation

Testing for common roots using the resultant or subtraction: (x^3 + 3x^2 + 4x + 5) - (x^3 + 2x^2 + 7x + 3) = x^2 - 3x + 2 = 0. Roots are x=1, x=2. Plugging these into the original equations shows neither is a root.

AI explanation

To find the common roots, we subtract the second equation from the first equation, which eliminates x cubed and gives x squared - 3x + 2 = 0. Factoring this quadratic equation yields (x - 1)(x - 2) = 0, so the only possible common roots are x = 1 and x = 2. Substituting x = 1 into the first original equation gives a sum of 13, not 0, and substituting x = 2 gives 29, not 0. Since neither value satisfies the original cubic equations, there are 0 common roots.

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