# Polynomial Inequalities ## Introduction A **polynomial inequality** involves a polynomial expression with an inequality sign ($<$, $>$, $\leq$, or $\geq$). **Examples:** - $x^2 - 5x + 6 > 0$ - $x^3 - 4x \leq 0$ - $2x^4 - 8x^2 + 6 \geq 0$ ## Solution Strategy The key steps for solving polynomial inequalities: 1. **Move all terms to one side** (get 0 on the other side) 2. **Factor the polynomial** completely 3. **Find critical values** (zeros/roots of the polynomial) 4. **Create a sign chart** to test intervals 5. **Determine solution intervals** based on the inequality 6. **Write the solution** in interval notation ## Sign Analysis Method ### Step-by-Step Process **Step 1: Set up the inequality** Rearrange so that the polynomial is on one side and 0 is on the other. **Step 2: Factor** Factor the polynomial completely to identify all zeros. **Step 3: Find critical values** Set each factor equal to zero and solve. These are the **critical values** that divide the number line into intervals. **Step 4: Create intervals** The critical values divide the number line into regions. Test a point from each region. **Step 5: Make a sign chart** Create a table showing: - The intervals - The sign of each factor in each interval - The overall sign of the product **Step 6: Identify solution intervals** - For $> 0$ or $\geq 0$: Choose intervals where the product is **positive** - For $< 0$ or $\leq 0$: Choose intervals where the product is **negative** - Use $\leq$ or $\geq$ to **include** critical values where the expression equals 0 - Use $<$ or $>$ to **exclude** critical values ## Sign Chart Template For $f(x) = (x - a)(x - b)(x - c)$ where $a < b < c$: | Interval | $(x-a)$ | $(x-b)$ | $(x-c)$ | $f(x)$ | |----------|---------|---------|---------|--------| | $x < a$ | $-$ | $-$ | $-$ | $-$ | | $a < x < b$ | $+$ | $-$ | $-$ | $+$ | | $b < x < c$ | $+$ | $+$ | $-$ | $-$ | | $x > c$ | $+$ | $+$ | $+$ | $+$ | ## Key Principles ### Sign of Linear Factors For $(x - a)$: - **Negative** when $x < a$ - **Zero** when $x = a$ - **Positive** when $x > a$ ### Multiplicity Effects - **Odd multiplicity**: Sign **changes** across the zero - **Even multiplicity**: Sign **stays the same** across the zero **Example:** $(x - 2)^2(x - 5)$ - At $x = 2$ (multiplicity 2): sign doesn't change - At $x = 5$ (multiplicity 1): sign changes ## Graphical Interpretation The solution to a polynomial inequality corresponds to where the graph satisfies the condition: - $f(x) > 0$: Graph is **above** the x-axis - $f(x) < 0$: Graph is **below** the x-axis - $f(x) \geq 0$: Graph is **above or on** the x-axis - $f(x) \leq 0$: Graph is **below or on** the x-axis ## Interval Notation **Symbols:** - $(a, b)$: Open interval (doesn't include endpoints) - $[a, b]$: Closed interval (includes both endpoints) - $[a, b)$: Half-open interval (includes $a$ but not $b$) - $(a, \infty)$: Unbounded interval extending to positive infinity **Union:** Use $\cup$ to combine disjoint intervals **Example:** $(-\infty, -2] \cup [3, \infty)$ ## Special Cases ### Always Positive or Always Negative Some polynomials never change sign: - $x^2 + 1 > 0$ for all real $x$ (always positive, never zero) - $-(x^2 + 4) < 0$ for all real $x$ (always negative) ### Includes Equality Remember: - $f(x) \geq 0$: Include values where $f(x) = 0$ (use brackets [ ]) - $f(x) > 0$: Exclude values where $f(x) = 0$ (use parentheses ( ))