Problem 38 - College Algebra

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. $$ x^{3}+7 x^{2}-x-7<0 $$

Step-by-Step Solution

Answer

The solution set on a real number line is \((-∞, -7) ∪ (-1, 1) ∪ (1, +∞)\).

Step 1: Find the roots of the polynomial

To find the roots of the polynomial \(x^{3} + 7x^{2} - x - 7 = 0\), we need to factorize it. This can be done using the rational root theorem or by synthetic division. For this polynomial, it can be factorized as \((x+1)(x-1)(x+7) = 0\). Solving this equation will give us the roots -1, 1 and -7.

Step 2: Set up the number line

We set up a number line using the roots of our polynomial. The number line has four intervals due to the three roots as follows: \(-\infty, -7\), \(-7, -1\), \(-1, 1\), and \(1, +\infty\).

Step 3: Test the intervals

For each of the intervals, choose a test point and substitute it in the inequality. If the value obtained is less than zero, then the entire interval satisfies the inequality. Otherwise, it doesn't. For this particular inequality, we find that the intervals \(-\infty, -7\), \(-1, 1\), and \(1, +\infty\) all satisfy the inequality.

Step 4: Write the solution in interval notation

The solution to the inequality in interval notation is \((-∞, -7) ∪ (-1, 1) ∪ (1, +∞)\). The interval notation includes the numbers that, when substituted into the inequality, make the inequality true.

Key Concepts

Polynomial Roots

When dealing with polynomial inequalities, the first step is to identify the polynomial roots. These are the solutions to the equation when it is set to zero. For a polynomial like \(x^3 + 7x^2 - x - 7 = 0\), finding these roots can help break down the inequality into manageable sections. Roots serve as critical points that split the real number line into intervals.

There are a few common methods to find polynomial roots. For instance, the rational root theorem can provide potential rational roots, while synthetic division can verify these roots efficiently. In our example, the polynomial can be factorized as \((x+1)(x-1)(x+7) = 0\). This leads to the roots: -1, 1, and -7.

• Root -1: Solving \(x + 1 = 0\)
• Root 1: Solving \(x - 1 = 0\)
• Root -7: Solving \(x + 7 = 0\)

These roots are crucial as they guide us in setting up intervals on the number line where the inequality will be analyzed.

Interval Notation

Interval notation offers a concise way of representing a range of values. It simplifies the depiction of solution sets for inequalities by specifying the start and end points of an interval.

An interval can be expressed in different ways:

• Open Interval: Uses parentheses, for instance, \((-1, 1)\), indicating that endpoints are not included.
• Closed Interval: Uses brackets, like \([-1, 1]\), signifying that endpoints are included.
• Infinite Intervals: Uses \((-\infty, a)\) or \((a, +\infty)\) to show unbounded ranges.

For the given polynomial inequality \(x^3 + 7x^2 - x - 7 < 0\), we are primarily using open intervals as the roots themselves do not satisfy the inequality. This leads to the solution given by the union of the intervals \((-∞, -7) \cup (-1, 1) \cup (1, +∞)\).

Number Line Graphing

To solve polynomial inequalities, using a number line graph is a helpful visual tool. It allows us to intuitively see where the inequality holds. You start by marking all the roots, which are \(-7\), \(-1\), and \(1\) for our polynomial.

The roots divide the number line into several intervals:

• \(-\infty, -7\)
• \(-7, -1\)
• \(-1, 1\)
• \(1, +\infty\)

Once the number line is set up, select a test point from each interval. Substituting these test points back into the inequality shows if the interval solves the inequality. For example, a test point \(x = -8\) confirms the inequality is true for \(-\infty, -7\). Similarly, checking for \(x = 0\) helps affirm it for \(-1, 1\).
Mark each satisfying interval on the number line graphically to understand the solution set. If the test point satisfies the inequality, shade that interval, leading to a clear visual of the solution set. This step is critical in understanding where the inequality holds across the entire real number line.