Solving Polynomials: Finding Roots Using a Table and Sign Changes

Identifying Exact Solutions

• 🎯 The video asks to identify the exact x-values where the polynomial function h(x) equals zero, using a provided table of values.
• ✅ From the table, h(x) is explicitly shown to be zero at x = -5 and x = 1.

Understanding Intermediate Solutions

• 💡 The concept of the Intermediate Value Theorem is applied to find intervals where solutions must exist.
• ⚠️ If a continuous function changes sign between two points (e.g., from positive to negative or negative to positive), it must cross the x-axis (h(x) = 0) at least once between those points.

Applying Sign Change Analysis

• 📈 The table shows h(x) values: h(-1) = 2, h(0) = -4. Since the function changes from positive to negative between x = -1 and x = 0, a solution to h(x) = 0 must exist in this interval.
• 🔍 This method helps locate intervals containing roots, even if the exact root isn't listed in the table.