Polynomial Sign Charts
Simple overview of creating sign charts for polynomials.
Sign Chart Core Concept
We use sign charts as a easy to follow method for finding when a function is positive or negative in a given interval
For a polynomial $f(x)$ when we have a $+$ signed interval on our sign chart then $f(x)>0$ in that interval, which tells us that the graph of our function will be above the $x$-axis.
Similarly, when a polynomial $f(x)$ has a $-$ signed interval on our sign chart then $f(x)<0$ in that interval, which tells us that the graph of our function will be below the $x$-axis.
The Process
To find a sign chart we do the following:
- Factor your polynomial completely to make the zeros easy to find.
- Find the zeros, the x-values that make the factors of our polynomial equal zero from step one. These are our $x$-intercepts.
- Put the zeros on a number line in order.
- Use those zeros to split the number line into intervals.
- Pick one test point from each interval.
- Plug each test point into the factors.
- Multiply the signs in each interval to get the sign of the polynomial.
After creating the sign chart we interpret what it means for an interval to have a $+$ or $-$ sign.
Compact Written Example
For a small written version of the same idea, suppose
\[f(x)=(x+3)(x-2)(x-5).\]The zeros are
\[x=-3, \quad x=2, \quad x=5.\]Those zeros split the number line into four intervals:
\[(-\infty,-3), \quad (-3,2), \quad (2,5), \quad (5,\infty).\]| Factor or product | $(-\infty,-3)$ | $(-3,2)$ | $(2,5)$ | $(5,\infty)$ |
|---|---|---|---|---|
| $x+3$ | $-$ | $+$ | $+$ | $+$ |
| $x-2$ | $-$ | $-$ | $+$ | $+$ |
| $x-5$ | $-$ | $-$ | $-$ | $+$ |
| $f(x)$ | $-$ | $+$ | $-$ | $+$ |
The final row (the row that starts with $f(x)$) is the only row that tells you whether the polynomial itself is above or below the $x$-axis.
So for this written example:
\[f(x)<0 \quad \text{when} \quad x \in (-\infty,-3) \cup (2,5)\]and
\[f(x)>0 \quad \text{when} \quad x \in (-3,2) \cup (5,\infty).\]Timestamp Guide
Next In The Series
After the sign chart is built, the next question is what the signs say about the graph. Use the companion guide for interpreting sign charts when you want to connect the chart to graph behavior.
Related JoeCMath clips: