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Pre-Calculus Polynomial and rational functions

Polynomial inequalities (interval notation)

20 practice questions 0 video lessons Theory + worked examples

Polynomial Inequalities

Common Core Pre-Calculus • Standard A-CED.1 • Polynomial & Rational Functions

Polynomial Inequalities is a topic in Polynomial & Rational Functions in the Common Core State Standards. It is aligned to Standard A-CED.1, which requires students to represent constraints by inequalities and express solutions in interval form.

A polynomial inequality asks where a polynomial is positive or negative, solved with a sign chart of its factored form and written in interval notation.

Common Core Pre-Calculus › Polynomial & Rational Functions › Polynomial Inequalities  —  Standard A-CED.1

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Theory

A polynomial inequality asks where a polynomial is positive or negative, e.g. \(p(x)>0\). Because a polynomial can only change sign at a zero, the roots split the number line into intervals on which the sign is constant.

The plan is a sign chart: mark the roots, then test one point in each interval to find the sign there.

Strict vs inclusive: for \(>\) or \(<\) the boundary roots are excluded (open); for \(\ge\) or \(\le\) they are included (closed).
Sign chart for a polynomial inequality The product (x minus 1)(x plus 2) is positive outside the roots negative 2 and 1, and negative between them. −2 1 + − +
Sign chart of \((x-1)(x+2)\): \(+\) outside the roots, \(-\) between them.
Where a polynomial is positive or negative The parabola (x minus 1)(x plus 2) lies below the x-axis between its roots negative 2 and 1, and above it outside. x y y<0 here
The same product is below the axis on \((-2,1)\) and above it outside.

The method rests on the sign-change rule:

\[p(x)\ \text{changes sign only at its zeros}\]
a polynomial changes sign only at its zeros

Write answers in interval notation, using \(\cup\) to join pieces and round vs square brackets for excluded vs included endpoints.

How to solve a polynomial inequality

  1. Move everything to one side so the other side is \(0\).
  2. Factor and find all real zeros (the boundary points).
  3. Mark them on a number line and test a point in each interval.
  4. Select the intervals with the sign you want; include or exclude endpoints per \(\le/\ge\) vs \(</>\).
  5. Write the solution in interval notation.
Example 1 — Product inequality
Solve \((x-1)(x+2)>0\).
Solution

The roots \(-2\) and \(1\) split the line into three intervals. Test a point in each.

\(x=-3\)\(\Rightarrow\)\((-)(-)=+\)
\(x=0\)\(\Rightarrow\)\((-)(+)=-\)
\(x=2\)\(\Rightarrow\)\((+)(+)=+\)

We want \(>0\): the outer intervals.

\[(-\infty,-2)\cup(1,\infty)\]
solution is x less than negative 2 or x greater than 1
Example 2 — Include the endpoints
Solve \(x^2-x-6\le 0\).
Solution

Factor, find the roots, and test signs. Because of \(\le\), the roots are included.

\((x-3)(x+2)\)\(\le\)\(0\)
\(\text{roots}\)\(=\)\(-2,\ 3\)

The product is \(\le 0\) between the roots:

\[[-2,\ 3]\]
solution is the closed interval from negative 2 to 3
Example 3 — A cubic inequality
Solve \(x(x-2)(x+3)\ge 0\).
Solution

Roots \(-3,0,2\) give four intervals; track the sign of the product.

\((-\infty,-3)\)\((-)(-)(-)=-\)
\((-3,0)\)\((-)(-)(+)=+\)
\((0,2)\)\((+)(-)(+)=-\)
\((2,\infty)\)\((+)(+)(+)=+\)

We want \(\ge 0\), including the roots:

\[[-3,\ 0]\cup[2,\ \infty)\]
solution is negative 3 to 0 together with 2 to infinity
Example 4 — A repeated factor
Solve \((x-1)^2(x+4)<0\).
Solution

The factor \((x-1)^2\) is never negative, so the sign is carried by \((x+4)\), except at \(x=1\) where the product is 0 (excluded by \(<\)).

\(x+4<0\)\(\Rightarrow\)\(x<-4\)

At \(x=1\) the product is 0, not \(<0\), so it is excluded.

\[(-\infty,\ -4)\]
solution is x less than negative 4

Common pitfalls

Get one side to zero first. You cannot read signs off \(p(x)>q(x)\) directly — move \(q\) over.
Repeated (even) factors don't change sign. \((x-1)^2\) stays \(\ge 0\); only odd-power factors flip the sign.
Endpoints depend on the symbol. Include roots for \(\le/\ge\), exclude them for \(</>\).

Frequently asked questions

How do you solve a polynomial inequality?

Move everything to one side, factor to find the zeros, build a sign chart by testing each interval, and select the intervals with the required sign.

Why do only the zeros matter?

A polynomial is continuous, so it can change from positive to negative only by passing through zero. Between consecutive zeros the sign is constant.

When are the endpoints included?

For \(\le\) or \(\ge\), include the boundary roots (square brackets). For strict \(<\) or \(>\), exclude them (round brackets).

What does a repeated factor do to the sign?

An even-power factor like \((x-1)^2\) never changes sign; the graph touches the axis there. Only odd-power factors flip the sign.

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