Question:
The cubic polynomial equation x³ + bx² + cx + d = 0 has
Last updated: 7/17/2022
![The cubic polynomial equation x³ + bx² + cx + d = 0 has](https://media.kunduz.com/media/sug-question/raw/54391632-1658050159.352591.jpeg?h=512)
The cubic polynomial equation x³ + bx² + cx + d = 0 has three roots X₁, X2, and X3. By expanding the product (x-x₁)(x - x₂)(x - x3), show that a) i) b = (x₁ + x₂ + x3); ii) c = x₁x₂ + x₁ x3 + x2X3; iii) d = -X₁X2X3. It is given that b = -9 and c = 45 for parts b) and c) below. b) i) In the case that the three roots x₁, x2, and x3 form an arithmetic sequence, show that one of the roots is 3. ii) Hence, determine the value of d. c) In another case the three roots form a geometric sequence. Determine the value of d.