1. Solving Polynomial Inequalities

2. Things to remember…
Use the Rational Root Theorem and the Quadratic Formula to solve a polynomial.
With inequalities, we can use a “test point” to determine the validity of a value.

3. Example 1: Solve 𝑥 3 +3 𝑥 2 −𝑥−3>0
Steps
Example
Set the polynomial equal to zero.
Solve the polynomial.
𝑥 3 +3 𝑥 2 −𝑥−3=0
𝑥={−3, 1, −1}

4. Example 1: Solve 𝑥 3 +3 𝑥 2 −𝑥−3>0
Use the solutions to break up the number line. -3 -1 1

5. Example 1: Solve 𝑥 3 +3 𝑥 2 −𝑥−3>0
Plug a “test point” from each section of the number in to the original inequality to test the validity of the value. -5 -3 -2 -1 1 2

6. Example 1: Solve 𝑥 3 +3 𝑥 2 −𝑥−3>0
Use the information from step 4 to write the solution to the inequality in interval notation.
−3, −1 𝑎𝑛𝑑 (1, ∞) -5 -3 -2 -1 1 2

7. Example 2: Solve x 4 −5 x 2 −36<0
Steps
Example
Set the polynomial equal to zero.
Solve the polynomial.
x 4 −5 x 2 −36=0
𝑥={−3, 3}

8. Example 2: Solve x 4 −5 x 2 −36<0
Use the solutions to break up the number line. -3 3

9. Example 2: Solve x 4 −5 x 2 −36<0
Plug a “test point” from each section of the number in to the original inequality to test the validity of the value. -3 3 -5 5

10. Example 2: Solve x 4 −5 x 2 −36<0
Use the information from step 4 to write the solution to the inequality in interval notation.
(−3, 3) -3 3 -5 5

11. Example 3: Solve 𝑥 4 −14 𝑥 2 ≤−45 Steps Example
Set the polynomial equal to zero.
Solve the polynomial.
𝑥 4 −14 𝑥 2 =−45
𝑥= −3, − 5 , 5 , 3

12. Example 3: Solve 𝑥 4 −14 𝑥 2 ≤−45
Use the solutions to break up the number line. -3
− 5 5 3

13. Example 3: Solve 𝑥 4 −14 𝑥 2 ≤−45
Plug a “test point” from each section of the number in to the original inequality to test the validity of the value. -3
− 5 5 3 −𝟓
−𝟐.𝟓 𝟎
𝟐.𝟓 𝟓

14. Example 3: Solve 𝑥 4 −14 𝑥 2 ≤−45
Use the information from step 4 to write the solution to the inequality in interval notation.
−3, − 𝑎𝑛𝑑 [ 5 , 3] -3
− 5 5 3 −𝟓
−𝟐.𝟓 𝟎
𝟐.𝟓 𝟓

15. You Try! Solve 𝑥 3 −2 𝑥 2 +6≥3𝑥