Polynomial Inequalities

Polynomial inequalities involve determining for which values of \( x \) the polynomial expression is positive, negative, or zero using the following inequlity signs

• Greater than : >
• Less than : <
• Greater than or equal to : ≥
• Less than or equal to : ≤

These inequalities determine the set of values (domain) for which the inequality holds true.

1. General Form of Polynomial Inequality

   let \(P(x)\) be a polynomial function. Then the inequality may be written as:
   • \(P(x)\) > 0
   • \(P(x)\) < 0
   • \(P(x)\) ≥ 0
   • \(P(x)\) ≤ 0

2. Solving Polynomial Inequalities

   Steps to solve:
   1. Set the inequality to zero: e.g. \(P(x) > 0 \)   ⇒ \(P(x) - 0 > 0 \)
   2. Find the roots (zeros) of the polynomial: Solve  \(P(x) 0 \)
   3. Divide the real number line into intervals using the roots.
   4. Test each interval to see if the inequality is satisfied.
   5. Write the solution in interval notation or as a set of inequalities.

3. Quadratic Inequalities (Degree 2)

   Example 1:

   Solve: \[x^2 - 5x + 6 \lt 0\] Step 1:
   Solve: \[x^2 - 5x + 6 = 0\] Factor: \[x^2 - 5x + 6 = 0\] \[(x - 2)(x -3) = 0\] \[\Rightarrow Roots: \quad 2,3\] Step 2: Intervals:
   • \( \left ( -\infty , 2 \right ) \)
   • (2, 3)
   • \() \left ( 3 ,+\infty \right ) \)
   step 3: Test sign of expression in each interval:
   • Pick \(x\) = 1 : ( 1 - 2)(1 - 3) = ( - 1)( - 2) > 0
   • Pick \(x\) = 2.5 : ( 2.5 - 2)(2.5 - 3) = ( 0.5)( - 0.5)< 0
   • Pick \(x\) = 4 : ( 4 - 2)(4 - 3) = ( 2)( 1) > 0
   Step 4: Inequality is Less than zero \Rightarrow solution is where the expression is Negative \[(2, 3 )\]

Quadratic Inequality Example

Solve the inequality: \( x^2 - 5x + 6 \leq 0 \)

Solution: \( 2 \leq x \leq 3 \)

1. Cubic Inequalities (Degree 3)

   Example 2:

   Solve: \[x^3 - x^2 - 6x \ge 0\] Step 1: Factor the expression: \[x^3 - x^2 - 6x = x(x^ - x - 6 ) = x (x - 3)( x + 2)\] Step 2: Roots: \[x \quad = \quad -2, \quad 0, \quad 3\] Step 3: Intervals:
   • \( \left ( -\infty , - 2 \right ) \)
   • (-2 , 0)
   • (0, 3)
   • \( \left (3, \infty \right ) \)
   Step 4: Sign Test:

   Interval	Test \(x\)	sign of \(x (x - 3)(x +2)\)	Result
   \( \left ( -\infty , - 2 \right ) \)	- 3	(-)(-)(-) = -	✖
   (-2 , 0)	-1	(-)(-)(+) = +	✔
   (0, 3)	1	(+)(-)(+) = -	✖
   \( \left (3, \infty \right ) \)	4	(+)(+)(+) = +	✔

   Step 5: Include points where expression is zero since it's "≥ 0": \[x = - 2, \quad 0, \quad 3 \] Final Solution: \[ [-2, 0 ] U [3, \infty ) \]

Cubic Inequality Example

Solve the inequality: \( x^3 - x^2 - 6x \geq 0 \)

Solution: \( x \leq -2 \cup 0 \leq x \leq 3 \)

1. Graphical Interpretation

   The graph of a polynomial can also help:
   • Where the graph is above the x-axis → \(P(x) > 0\)
   • Where the graph is on the x-axis → \(P(x) = 0\)
   • Where the graph is below the x-axis → \(P(x) < 0\)

2. Tips and Considerations

   • for even-degree polynomials,end behavior is in the same direction.
   • For odd-degree polynomials, ends go in opposite directions.
   • Multiplicity of roots:
     • Odd multiplicity → sign changes
     • Even multiplicity → sign does not change