Solving Polynomial Inequalities

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.

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}

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

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

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

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}

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

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

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

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

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

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

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, − 5 𝑎𝑛𝑑 [ 5 , 3] -3 − 5 5 3 −𝟓 −𝟐.𝟓 𝟎 𝟐.𝟓 𝟓

You Try! Solve 𝑥 3 −2 𝑥 2 +6≥3𝑥