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CA Foundation · Quantitative Aptitude · Equations

What is the product of all the roots of the equation x³ − 4x² + x + 6 = 0, and how many of its roots are integers?

The product of the roots is −6 and all three roots are integers. The roots are −1, 2 and 3, found by testing x = −1 and factorising the remaining quadratic. Their product −6 matches −d/a for the cubic.

  1. AProduct 6; three integer roots
  2. BProduct −6; three integer rootsCorrect
  3. CProduct −6; one integer root
  4. DProduct 6; one integer root

Explanation

For a monic cubic the product of the roots is −d = −6. Testing x = −1: −1 − 4 − 1 + 6 = 0, so it is a root. Dividing gives x² − 5x + 6 = (x − 2)(x − 3). Roots are −1, 2, 3, with product −6, all integers. Product 6 would ignore the sign rule.

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