91Ó°ÊÓ

First, consider the polynomial function \(p(x) = x^{4} + 7x^{3} - 9\). It's a continuous function for all real numbers since it's a polynomial. This understanding will be fundamental for applying the intermediate value theorem.

Proof for existence of the two roots can be given by applying the Intermediate Value Theorem (IVT). IVT states that if a function \(f\) is continuous on the interval \([a, b]\) and \(N\) is any number between \(f(a)\) and \(f(b)\), then there exists a number \(c\) in the interval \((a, b)\) for which \(f(c) = N\). We can apply this theorem by choosing suitable points \(a\) and \(b\), such that \(p(a)\) and \(p(b)\) have different signs. For example, let's choose \(a = -10\) and \(b = 0\). Calculating respective \(p(a)\) and \(p(b)\), we can see that \(p(-10) < 0\) and \(p(0) > 0\). Since \(p(x)\) is continuous, by the Intermediate Value Theorem, there must be at least one real root in the interval (-10,0). The same way we can find another interval, for instance (0,1), where \(p(0) > 0\) and \(p(1) < 0\), so that there is at least a second root. Hence, the function \(p(x) = x^{4} + 7x^{3} - 9\) has at-least two real roots.

The actual roots can be located using numerical methods. A simple approach, which demonstrates the concept, is the binary search method. For each interval identified in step 2, the method starts by calculating the middle point c = (a+b)/2. If \(p(c)\) is very close to zero, we can consider it as a root. If \(p(c)\) is not close to zero, and \(p(c)\) has different sign with \(p(a)\), we update \(b\) to \(c\), else we update \(a\) to \(c\), and repeat the process until the desired precision is reached.

By applying the method defined in Step 3 to the two intervals (-10,0) and (0,1) and using a calculator, we can locate the two roots approximately to two decimal places: -6.90 and 0.13.