If a > 0 and all four roots are positive, what can be said about the sign of b?

If a > 0 and all four roots are positive, what can be said about the sign of b?

• A. b > 0
• B. b < 0
• C. b = 0
• D. b can be any sign

Answer: (B)

Think of the polynomial as a(x − r1)(x − r2)(x − r3)(x − r4) with a > 0 and all roots r1, r2, r3, r4 positive. When you expand, the coefficient of x^3 is −a times the sum r1 + r2 + r3 + r4. Since each root is positive, that sum is positive, and with a > 0 the whole thing is negative. So the sign of the x^3 coefficient, namely b, must be negative. Therefore b < 0. The constant term would be a times the product r1r2r3r4, which is also positive, and the other signs follow the same pattern.

Think of the polynomial as a(x − r1)(x − r2)(x − r3)(x − r4) with a > 0 and all roots r1, r2, r3, r4 positive. When you expand, the coefficient of x^3 is −a times the sum r1 + r2 + r3 + r4. Since each root is positive, that sum is positive, and with a > 0 the whole thing is negative. So the sign of the x^3 coefficient, namely b, must be negative. Therefore b < 0. The constant term would be a times the product r1r2r3r4, which is also positive, and the other signs follow the same pattern.