To solve this week's POTW, you need to first set the given expression, (x+1)(x+2)(x+5)≥36x, equal to 0. Then, you can use the quadratic formula or factorization to find the roots of the equation. Next, you can plot the roots on a number line and test different intervals to determine the solution set. Finally, you can write the final solution in interval notation.
Setting the expression equal to 0 allows you to find the roots of the equation, which are the values of x that make the expression equal to 0. These roots are important in determining the intervals to test for the solution set.
The quadratic formula is used to find the roots of a quadratic equation in the form ax^2 + bx + c = 0. In this case, the given expression can be rewritten as x^3 + 8x^2 + 17x + 10 = 0. Then, you can plug in the values of a, b, and c into the quadratic formula, x = (-b ± √(b^2 - 4ac)) / 2a, to find the two roots of the equation.
Factorization is the process of writing an expression as a product of its factors. In this case, the given expression can be factored into (x+1)(x+2)(x+5) = 0. By setting each factor equal to 0, you can find the three roots of the equation. This method is often quicker and easier than using the quadratic formula.
After finding the three roots of the equation, you can plot them on a number line and test different intervals to determine the solution set. The solution set can then be written in interval notation, using parentheses for open intervals and brackets for closed intervals. For example, if the solution set is x ≤ -5 or -2 ≤ x ≤ 2, it can be written as (-∞, -5] ∪ [-2, 2].