Pre-Calculus: Solving Polynomial Inequalities

Solving polynomial inequalities is a foundational skill that bridges algebra with calculus and has direct applications in engineering, economics, and data science. It answers critical questions: When is a projectile above a certain height? For what production levels is profit positive? Mastering this technique requires moving beyond solving equations to analyzing sign behavior across entire intervals of the real number line.

From Equation to Inequality: The Foundational Shift

The core task is to determine where a polynomial expression, such as $p(x)=2x^3-3x^2-11x+6$, is greater than, less than, greater than or equal to, or less than or equal to zero. The crucial shift from solving an equation $p(x)=0$ is that the solution to an inequality is never just a set of points—it is one or more continuous intervals. The zeros of the polynomial (the solutions to $p(x)=0$) are not typically the answer themselves; instead, they serve as critical boundary points that partition the number line into testable regions. The process hinges on a key property of polynomial functions: they are continuous and can only change sign at their real zeros.

The Step-by-Step Algorithm: Factoring, Zeros, and the Sign Chart

A reliable, systematic method is essential for solving any polynomial inequality. Follow these steps.

Step 1: Arrange and Factor. First, move all terms to one side of the inequality, leaving zero on the other side. Then, completely factor the polynomial. For our example, to solve $2x^3-3x^2-11x+6\ge 0$, we first have zero on the right. Factoring yields $(2x-1)(x+2)(x-3)\ge 0$.

Step 2: Identify Critical Numbers. The critical numbers are all real zeros of the polynomial (from the factors) and any values that make the expression undefined (not applicable for polynomials). From our factored form $(2x-1)(x+2)(x-3)=0$, the zeros are $x=\frac{1}{2}$, $x=-2$, and $x=3$. These three numbers will divide our number line.

Step 3: Construct a Sign Chart. Plot your critical numbers in order on a number line. This creates intervals: $(-\infty ,-2)$, $(-2,\frac{1}{2})$, $(\frac{1}{2},3)$, and $(3,\infty )$. The sign chart is a tool to determine the sign (positive or negative) of the entire polynomial within each interval. Since the polynomial is a product of factors, its sign is the product of the signs of each individual factor in that interval.

Step 4: Test Each Interval. Choose a simple test number from each interval and evaluate the sign of each factor, then the overall product.

• Interval $(-\infty ,-2)$: Test $x=-3$. Signs: $(2(-3)-1)=(-)$, $(-3+2)=(-)$, $(-3-3)=(-)$. Product: $(-)\ast (-)\ast (-)=(-)$.
• Interval $(-2,\frac{1}{2})$: Test $x=0$. Signs: $(-)$, $(+)$, $(-)$. Product: $(-)\ast (+)\ast (-)=(+)$.
• Interval $(\frac{1}{2},3)$: Test $x=1$. Signs: $(+)$, $(+)$, $(-)$. Product: $(+)\ast (+)\ast (-)=(-)$.
• Interval $(3,\infty )$: Test $x=4$. Signs: $(+)$, $(+)$, $(+)$. Product: $(+)\ast (+)\ast (+)=(+)$.

Step 5: State the Solution. We want where $p(x)\ge 0$ (positive or zero). From the chart, the polynomial is positive on $(-2,\frac{1}{2})$ and $(3,\infty )$. It equals zero at $x=-2,\frac{1}{2},3$. For "greater than or equal to," we include the endpoints where the polynomial is zero. Therefore, the solution in interval notation is $[-2,\frac{1}{2}]\cup [3,\infty )$. For a strict inequality like $>$, we would use parentheses at those endpoints: $(-2,\frac{1}{2})\cup (3,\infty )$.

The Nuance of Repeated Roots and Multiplicity

A polynomial may have a repeated root, such as $(x-1{)}^2$ or $(x+4{)}^3$. The exponent is the multiplicity of the root. Multiplicity critically affects the graph's behavior at the x-axis but also has a subtle effect on inequalities. The key rule: The polynomial will not change sign at a root of even multiplicity. It will touch the x-axis and turn around, remaining on the same side.

For example, solve $(x-4{)}^2(x+1)>0$. The zeros are $x=4$ (multiplicity 2, even) and $x=-1$ (multiplicity 1, odd). Construct the sign chart with intervals $(-\infty ,-1)$, $(-1,4)$, and $(4,\infty )$.

• For $x=0$ in $(-1,4)$: $(0-4{)}^2=(+)$, $(0+1)=(+)$. Product is positive.
• For $x=5$ in $(4,\infty )$: $(+)\ast (+)$ = positive.

Notice the sign is positive on both sides of $x=4$ because of the even power. The polynomial changes sign only at $x=-1$ (odd multiplicity). The solution to $>0$ is $(-\infty ,-1)\cup (-1,4)\cup (4,\infty )$. Importantly, $x=4$ is not included because the polynomial is zero there, and we have a strict inequality ($>$). If the inequality was $\ge 0$, the solution would be all real numbers, as the expression is positive everywhere except zero at $x=4$ and $x=-1$.

The Graphical Connection: Regions Above or Below the X-Axis

Every algebraic solution corresponds directly to a visual one. Solving $p(x)>0$ means finding the x-values for which the graph of $y=p(x)$ is above the x-axis. Solving $p(x)<0$ means finding where the graph is below the x-axis. The zeros are the x-intercepts. Your sign chart is essentially a one-dimensional analysis of the graph's vertical position. This connection is vital for calculus, where you'll use the derivative $p^{\prime }(x)$ to find where a function is increasing or decreasing, which involves solving different inequalities.

Common Pitfalls

1. Incorrectly Handling Endpoints: The most frequent error is misusing brackets and parentheses in interval notation. Remember: Use brackets $[]$ only if the endpoint is included. This happens when you have $\le$ or $\ge$ and the polynomial is defined and equal to zero at that point (it is not a vertical asymptote, which doesn't occur with pure polynomials). For strict inequalities ($<$ or $>$), always use parentheses $()$, even if the number is a zero.

1. Misreading the Inequality Direction: After constructing your sign chart, always circle the intervals that match your original inequality. A positive sign chart region corresponds to $>0$. If your initial problem was $p(x)<0$, you select the negative intervals. Double-check this step.

1. Forgetting to Factor Completely: You cannot determine correct critical numbers or signs if the polynomial is not factored to its simplest polynomial factors. Always factor out the Greatest Common Factor (GCF) first, then proceed with grouping or other methods. An unfactored form hides the zeros.

1. Ignoring Multiplicity with Even Powers: When a factor has an even exponent, the expression does not change sign at that root. Students often mistakenly split the interval there, testing on either side and getting the same sign, which should confirm the rule, not cause confusion. Mark even-multiplicity roots on your sign chart with a special note (e.g., "NO SIGN CHANGE").

Summary

• The solution to a polynomial inequality is one or more intervals, found by using the zeros of the polynomial to partition the number line.
• The sign chart is a systematic, foolproof tool for determining the sign of the polynomial in each interval by testing a sample point.
• Always express your final answer in interval notation, carefully choosing brackets or parentheses based on the inequality symbol and whether the endpoint is a valid, included zero.
• The multiplicity of a zero matters: at roots with odd multiplicity, the graph and the sign of the polynomial cross the axis (change sign); at roots with even multiplicity, they touch and turn back (no sign change).
• Algebraically solving $p(x)>0$ or $p(x)<0$ is equivalent to graphically finding where the curve $y=p(x)$ is above or below the x-axis, respectively.