Problem 4

Explanation

We will simplify the forms of $P$ and $Q$ first to compare their values.

Simplifying P Value

Given $P = \frac{1 - x^6}{1 - x^3}$. Recall the difference of squares $a^2 - b^2 = (a - b)(a + b)$. We can view $1 - x^6$ as $1^2 - (x^3)^2$.

Simplifying Q Value

Given $Q = (1 + x)^3$. We expand it using binomial expansion:

Comparing P and Q

We find the difference between $Q$ and $P$:

Given the condition $0 < x < 1$. Since $x$ is positive ($x > 0$), then:

• $3x > 0$
• $3x^2 > 0$

Thus, their sum must be positive:

Or it can be written as $P < Q$.

Conclusion

Thus, the correct relationship is $P < Q$.