Multiple choice

If $1, 2, 3$ are the roots of the equation $x^{4} + ax^{2} + bx + c = 0$ then the value of $c$ is :

  1. $18$
  2. $-36$
  3. $30$
  4. $32$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If 1, 2, 3 are roots of x^4 + ax^2 + bx + c = 0, there must be a fourth root, say r. The polynomial is (x-1)(x-2)(x-3)(x-r) = 0. Expanding this, the constant term c is the product of the roots: 1 * 2 * 3 * r = 6r. However, the x^3 coefficient must be 0. The sum of roots is 1+2+3+r = 6+r = 0, so r = -6. Thus c = 1*2*3*(-6) = -36.

AI explanation

Let the fourth root of the polynomial be r. Since the leading coefficient is 1 and there is no x^3 term, the sum of all four roots must equal zero, meaning 1 + 2 + 3 + r = 0, so r = -6. The constant term c represents the product of all the roots, but factoring in the leading coefficient sign for an even degree polynomial yields c = (1)(2)(3)(-6). Multiplying these values gives c = -36.

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