Question

Let $f(x)=3x^6 - 2x^2 - 8$. Which of the following statements is (are) true?

• A. The sum of all its roots is zero.
• B. The product of its roots is $-\dfrac{8}{3}$.
• C. The sum of all its roots is $\dfrac{2}{3}$.
• D. Complex roots are conjugates of each other.

Hint:

For real-coefficient polynomials, non-real roots must appear in conjugate pairs.

Answer 1

The Correct Option is A, B, D

Step 1: Use general polynomial identity.
For $3x^6 - 2x^2 - 8$, there is no $x^5$ term. Thus the sum of all roots = 0 (true).

Step 2: Product of roots.
Product = $(-1)^6 \dfrac{-8}{3} = -\dfrac{8}{3}$ → matches (B), but note that sign is correct only if all roots counted (yes). So (B) is also correct mathematically.

Step 3: Complex-root property.
Real coefficients → complex roots occur in conjugate pairs ⇒ (D) true.

Step 4: Conclusion.
(A) and (D) are always true from polynomial structure.