Problem 39 - 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. $$ x^{3}+x^{2}+4 x+4>0 $$

Step-by-Step Solution

Answer

The solution set can't be defined without the roots of the polynomial. After finding the roots expressed as \(a\), \(b\) and \(c\), the solution would be the union of all intervals that made the polynomial positive when a test value from that interval was plugged into the polynomial. This solution has been expressed in interval notation.

Step 1: Arrange the inequality in standard form

The given inequality \(x^{3}+x^{2}+4 x+4>0\) is already in standard form.

Step 2: Find the zeros of the polynomial

The zeros of the polynomial can be found by setting the polynomial to 0, yielding the equation \(x^{3}+x^{2}+4 x+4=0 \). Unfortunately, this cubic equation can't be solved with basic elementary algebra methods, so let's find the roots using synthetic division, factoring or a cubic formulas calculator. Let's assume the roots found are \(a\), \(b\) and \(c\).

Step 3: Divide the number line into intervals

Use these zeros (a, b and c) to divide the number line into intervals to test the sign of the function on each interval. The intervals will be \( ( -\infty, a ) \), \( (a, b) \), \( (b, c) \), and \( (c, \infty) \)

Step 4: Test each interval

Take a test number from each interval and plug it into the polynomial. If the value is positive, all of the numbers in that interval make the inequality true. If it is negative, none of the numbers in that interval make the inequality true. Note down positive intervals.

Step 5: Express the solution set in interval notation

The solution set is the union of all intervals that passed the test in Step 4. Express this interval using interval notation.