Problem 102 - Algebra and Trigonometry

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 \((x+3)(x-5) \leq 0\).

Step 1: Express the range as an inequality

Start the process by expressing the range (-3 to 5) as an inequality. It would be written as \(-3 \leq x \leq 5\) as it includes both -3 and 5.

Step 2: Formulate the polynomial inequality

To convert this inequality into a polynomial inequality, you could use the expression \((x+3)(x-5)\). When expanded this gives \(x^2 -2x -15\). Moreover, because it should hold for the values between and including -3 and 5, the inequality should be \((x+3)(x-5) \leq 0\). That ensures that the values within that range will give a product that is less than or equal to zero.

Step 3: Verifying the solution

The validity of the solution could be verified by substituting a number between -3 and 5 into the inequality. If the inequality holds true, then the solution is correct. For example, if x is 0, then the left-hand side of the inequality becomes \((-3)(-5)=15\) which is indeed bigger than 0, so the inequality holds true.