Suppose that a real quartic degree 4) polynomial has a positive discriminant. Wh

Question

Suppose that a real quartic degree 4) polynomial has a positive discriminant. What can you say about the number of real roots?

Explanation / Answer

A zero discriminant means that there is a repeated root. Now suppose your coefficients are all real numbers, and there happen to be four real roots (p, q, r, s). Then the discriminant is [(p-q)(p-r)(p-s)(q-r)(q-s)(r-s)]^2 > 0. If you have two real roots and two complex roots (p, q, r+si, r-si), then the discriminant is (p-q)^2 [(p-r-si)(p-r+si)]^2 [(q-r-si)(q-r+si)]^2 (2si)^2 = (p-q)^2 ((p-r)^2 + s^2)^2 ((q-r)^2 + s^2)^2 * (-4s^2) < 0. Similarly, if you have four complex roots (p+qi, p-qi, r+si, r-si), then the discriminant will be positive. So you may conclude that if the discriminant is negative, then you have two real roots and two complex roots. If the discriminant is positive, it could be four real roots or four complex roots.

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.