Final answer:
The three roots of the equation x³ - 3x² - 4x + 12 = 0, given that two of the roots are equal but opposite in signs, are p, -p, and 3.
Step-by-step explanation:
Given that two of the roots are equal but opposite in signs, we can start by assuming that the two equal roots are p and -p. We need to find the third root.
By the Vieta's formulas, the sum of the roots of the equation is 3, the coefficient of the x³ term, divided by the coefficient of the x³ term, which is 1.
So, p + (-p) + r = 3, where r is the third root. Solving for r, we get r = 3.
Therefore, the three roots of the equation x³ - 3x² - 4x + 12 = 0, given that two of the roots are equal but opposite in signs, are p, -p, and 3.
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