Multiple choice

$x^{2}+x+1$ is a factor of $ax^{3}+bx^{2}+cx+d=0$ where $(a,b,c,d, \in \ R)$, then the real root of above equation is

  1. $-d/a$
  2. $d/a$
  3. $(b-a)/a$
  4. $(a-b)/a$
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A Correct answer
Explanation

Since x^2 + x + 1 is a factor of ax^3 + bx^2 + cx + d = 0, we can write the cubic as (x^2 + x + 1)(ax + k) = 0. Expanding this gives ax^3 + (a+k)x^2 + (a+k)x + k = 0. Comparing the constant term, k = d. The real root comes from the linear factor ax + d = 0, which gives x = -d/a.

AI explanation

Since x^2 + x + 1 is a factor of ax^3 + bx^2 + cx + d = 0, we can write the cubic as a(x + 1)(x^2 + x + 1) = 0. Expanding this product gives ax^3 + 2ax^2 + 2ax + a = 0. Comparing the constant terms reveals that d = a. The real root comes from the linear factor x + 1 = 0, meaning x = -1. Substituting d = a into the expression -d/a gives -1, establishing the real root as -d/a.

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