Multiple choice

Directions: Answer the question independently. What may be true about the roots of the equation ax3 + bx - 1 = 0?

  1. It has two irrational and one rational root.

  2. It has one irrational and two rational roots.

  3. All its roots are complex.

  4. None of the above.
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A Correct answer
AI explanation

A cubic polynomial with rational coefficients must have roots that are either all rational or include at least one irrational root and one rational root, because irrational roots of rational polynomials must occur in conjugate pairs if they are quadratic surds. By testing small rational values, we see that x = 1 is a root because a(1) cubed plus b(1) minus 1 equals zero, meaning the polynomial is divisible by (x - 1). Dividing the cubic polynomial by (x - 1) yields the quadratic factor ax squared plus ax plus (b plus a), whose discriminant is b squared plus a squared minus 2ab plus 4a squared. Unless a equals b, this discriminant is not a perfect square, meaning the remaining two roots are irrational. Thus, the equation has exactly two irrational roots and one rational root.

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