Multiple choice

lf $a,b, \mathrm{g}$ are the roots of the equation $x^{3}+ax^{2}+bx+c=0$, then ${a}^{-1}+{b}^{-1}+{g}^{-1}{=}$

  1. $\displaystyle \dfrac{a}{c}$
  2. $-\displaystyle \dfrac{b}{c}$
  3. $\displaystyle \dfrac{c}{a}$
  4. $\displaystyle \dfrac{b}{a}$
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B Correct answer
Explanation

For roots a, b, and g, the product of the roots is -c by Vieta's formulas. Therefore, 1/a + 1/b + 1/g = (ab + bg + ga)/(abg) = -b/c.

AI explanation

Let the roots of the cubic equation be l, m, and n. By Vieta's formulas, the sum of the roots taken two at a time is lm + ln + mn = b, and the product of the roots is lmn = -c. To find the sum of their reciprocals, compute 1/l + 1/m + 1/n. Finding a common denominator gives (lm + ln + mn) / lmn. Substituting the Vieta's formulas results gives b / -c, which equals -b/c.

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