Problem 102 - College Algebra

Question

Write a polynomial inequality whose solution set is \([-3,5]\)

Step-by-Step Solution

Answer

The polynomial inequality whose solution set is \([-3,5]\) is \( -1 * (x + 3)(x - 5) \leq 0\).

Step 1: Identify the required roots

First, we notice that we want the solutions of the inequality to lie in the interval \([-3,5]\), including the end points. This means that we want the polynomial to be zero at \(x = -3\) and \(x = 5\). In other words, \(x + 3\) and \(x - 5\) are factors of the polynomial.

Step 2: Construct the polynomial

Remember that a polynomial will be zero whenever its factors are zero. So our polynomial should be some multiple of \((x + 3)(x - 5)\). Furthermore, since we want the polynomial to be less than or equal to zero in the interval, we should use the negative of this product to make sure we get a downward-opening parabola. So our polynomial is \(-1 * (x + 3)(x - 5)\).

Step 3: Convert to inequality

The polynomial is now created, but the task was to create a polynomial inequality. As the polynomial value is required to be less than or equal to zero, the inequality is \( -1 * (x + 3)(x - 5) \leq 0\). This is the polynomial inequality sought.