To determine the number of real solutions for the equation x^7 + 14x^5 + 16x^3 + 30x - 560 = 0, we can use the concept of the Intermediate Value Theorem.
The Intermediate Value Theorem states that if a continuous function takes on two different values at two points, then it must also take on every value in between those two points.
In this case, the equation is a polynomial function of degree 7. Since polynomial functions are continuous, we can apply the Intermediate Value Theorem to determine the number of real solutions.
By plugging in some values for x, we can find that the function takes on both positive and negative values. For example:
- When x = -10, the function value is negative.
- When x = 0, the function value is negative.
- When x = 10, the function value is positive.
Since the function changes sign between x = -10 and x = 0 and also between x = 0 and x = 10, by the Intermediate Value Theorem, there must be at least one real solution in each of these intervals.
Therefore, the equation x^7 + 14x^5 + 16x^3 + 30x - 560 = 0 has at least 2 real solutions.
Since the degree of the polynomial is 7, it can have at most 7 real solutions (counting multiplicities).
However, the correct answer is A) 1, which means that there is only one real solution. This suggests that there are multiple complex solutions (which are not real).
Therefore, the equation x^7 + 14x^5 + 16x^3 + 30x - 560 = 0 has 1 real solution.