D = \{x \in \mathbb{R} \mid (x^2 - 10x + 21)(x^3 - x) = 0\}
Answer (1)
To solve the given problem, we need to find the values of x that satisfy the equation ( x 2 − 10 x + 21 ) ( x 3 − x ) = 0 . This involves solving the equation by setting each factor to zero, since if a product of factors equals zero, at least one of the factors must be zero.
Solve the first factor: x 2 − 10 x + 21 = 0 This is a quadratic equation, which can be factored as: ( x − 3 ) ( x − 7 ) = 0 Thus, x = 3 or x = 7 .
Solve the second factor: x 3 − x = 0 Factor out an x : x ( x 2 − 1 ) = 0 Set each factor to zero:
x = 0
Solve the quadratic x 2 − 1 = 0 : ( x − 1 ) ( x + 1 ) = 0 So, x = 1 or x = − 1 .
By combining all the solutions from these factors, we find that the set D is: D = { − 1 , 0 , 1 , 3 , 7 }
These are the values of x that make the original equation true, where either x 2 − 10 x + 21 = 0 or x 3 − x = 0 . Therefore, the solution to the problem is the set of real numbers − 1 , 0 , 1 , 3 , and 7 .
