Question 40

The equation has exactly three distinct real roots.

The equation (x-1)(x²+x+1)(x²-3x-4)=0 has three distinct real roots. This is determined by analyzing each factor of the equation.

A) One

This option is incorrect because the equation has more than one factor that can yield real roots. Specifically, the factor (x-1) contributes one real root, while the other factors provide additional roots.

B) Two

This option is also incorrect. While it is true that two roots could be a possibility if only one of the quadratic factors contributed real roots, the presence of the linear factor (x-1) and the other quadratic factor (x²-3x-4) ensures more than two distinct real roots.

C) Three

This option is correct because the factor (x-1) provides one distinct root, and the quadratic factor (x²-3x-4) can be factored further into (x-4)(x+1), which provides two additional distinct roots. The factor (x²+x+1) has no real roots as its discriminant is negative. Therefore, the total number of distinct real roots is three.

D) Four

This option is incorrect. The equation cannot have four distinct real roots because the quadratic factor (x²+x+1) does not contribute any real roots. The maximum number of distinct real roots is limited by the existing factors, which total to three.

E) Five

This option is incorrect as well, as it exceeds the maximum number of real roots possible based on the factors present in the equation. The equation can only yield up to three distinct real roots.

Conclusion

The analysis confirms that the equation has exactly three distinct real roots, coming from the combination of the linear factor and one of the quadratic factors. All other options fail to account for the contribution of these factors accurately, leading to an underestimation or overestimation of the number of distinct real roots.