Problem 37 - Algebra and Trigonometry

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x^{3}-3 x^{2}-9 x+27<0 $$

Step-by-Step Solution

Answer

Without specifying the given roots, the short answer can be generally put as 'The solution set of the inequality is the union of intervals on which the polynomial is negative, and it is expressed in interval notation and graphed on a number line.'

Step 1: Find the Roots of the Polynomial

To find the roots of the polynomial, set the polynomial equal to zero and solve for \(x\). The equation becomes: \(x^{3}-3x^{2}-9x+27 = 0\). Solve this equation to find the roots of \(x\).

Step 2: Determine the Intervals

After finding the roots of the polynomial, these roots will divide the number line into several intervals. The polynomial changes its sign at each root. Thus, the solution of the inequality will be a union of some of these intervals.

Step 3: Evaluate the Sign of the Polynomial on Each Interval

Pick a test point from each interval, and substitute it into the polynomial to check whether the polynomial is positive or negative on this interval. If it holds true with the inequality, then all the numbers in this interval are the solutions.

Step 4: Express the Solution in Interval Notation

After finding the intervals for which the inequality holds true, write these intervals in interval notation. The interval notation \( (a, b) \) represents all numbers between \(a\) and \(b\). If \(a\) or \(b\) is included in the interval, use square brackets [ or ] instead of round brackets ( or ).

Step 5: Draw the Solution Set on a Number Line

Finally, represent the solution set on a real number line. Plot the roots on the number line, and draw a line or a ray for each interval of the solution. If a root is included in the solution, the corresponding point on the number line is filled, otherwise, it is unfilled.