The equation (x-1)(x²+x+1)(x²-3x-4)=0 has exactly how many distinct real roots?

Rationale

The equation (x-1)(x²+x+1)(x²-3x-4)=0 has exactly three distinct real roots.

To determine the number of distinct real roots in the given equation, we analyze each factor separately. The first factor, (x-1), provides one real root at x = 1. The second factor, (x²+x+1), has no real roots since its discriminant is negative. The third factor, (x²-3x-4), can be factored to (x-4)(x+1), yielding two additional distinct real roots at x = 4 and x = -1. Combining these, we find a total of three distinct real roots.

A (One) YOUR ANSWER

This option suggests there is only one distinct real root. However, since the factor (x²-3x-4) contributes two additional roots and (x-1) contributes one, it is incorrect to claim only one root exists.

B (Two) YOUR ANSWER

This choice indicates two distinct real roots. While the factor (x-1) does provide one root, the (x²-3x-4) factor contributes two more distinct roots, leading to a total of three, making this choice incorrect.

C (Three) CORRECT

This option is correct as it counts the one real root from (x-1) and the two distinct real roots from (x²-3x-4). Thus, the total number of distinct real roots is indeed three.

D (Four) YOUR ANSWER

This choice incorrectly suggests there are four distinct real roots. The equation only produces three distinct real roots, as the factor (x²+x+1) does not contribute any real roots.

E (Five) YOUR ANSWER

This option suggests five distinct real roots, which is not possible given the factors. The total number of distinct real roots is limited to three based on the analysis of all factors.

Conclusion

In conclusion, the equation (x-1)(x²+x+1)(x²-3x-4)=0 yields exactly three distinct real roots, derived from the contributions of the factors involved. The analysis highlights that while one factor provides a root, others can either contribute additional roots or none at all, leading to the final count of three distinct real roots.

The equation (x-1)(x²+x+1)(x²-3x-4)=0 has exactly how many distinct real roots?

The equation (x-1)(x²+x+1)(x²-3x-4)=0 has exactly three distinct real roots.

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