Solve each polynomial inequality. Give the solution set in interval notation. (x - 3)(x - 4)(x - 5)² ≤ 0

Verified step by step guidance

1. First, identify the critical points by setting each factor equal to zero: solve \(x - 3 = 0\), \(x - 4 = 0\), and \(x - 5 = 0\). These points divide the number line into intervals.
2. Next, determine the multiplicity of each root: \(x = 3\) and \(x = 4\) have multiplicity 1 (since their factors are to the first power), and \(x = 5\) has multiplicity 2 (since \((x - 5)^2\) is squared).
3. Then, choose test points from each interval created by the critical points (for example, values less than 3, between 3 and 4, between 4 and 5, and greater than 5) and substitute them into the expression \((x - 3)(x - 4)(x - 5)^2\) to determine the sign (positive or negative) of the product in each interval.
4. Analyze the sign of the product on each interval, remembering that factors raised to an even power (like \((x - 5)^2\)) are always non-negative, which affects the overall sign of the product.
5. Finally, combine the intervals where the product is less than or equal to zero (\(\leq 0\)) and include the critical points where the expression equals zero, then express the solution set in interval notation.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality signs (>, <, ≥, ≤). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Critical Points and Sign Analysis

Critical points are the roots of the polynomial where the expression equals zero. These points divide the number line into intervals. By testing values in each interval, you determine whether the polynomial is positive or negative there, which helps identify where the inequality holds.

Multiplicity of Roots

The multiplicity of a root refers to how many times a factor appears in the polynomial. Even multiplicities cause the graph to touch the x-axis and turn around, not crossing it, while odd multiplicities cause the graph to cross the axis. This affects the sign changes around critical points.