Polynomial Sign Charts
This page follows the JoeCMath video Master Polynomial Sign Charts in 8 Minutes. The video walks through the process of building a sign chart for a polynomial, then using the final row to decide where the function is positive or negative.
Main Idea
A sign chart is a compact way to keep track of whether each factor is positive or negative on each interval.
For a polynomial $f(x)$, the final row of the sign chart answers questions like:
\[f(x)>0\]and
\[f(x)<0.\]When $f(x)>0$, the graph is above the $x$-axis. When $f(x)<0$, the graph is below the $x$-axis.
The Process
The video builds the sign chart one layer at a time:
- Start with the polynomial used in the video.
- Find the zeros, which are also the $x$-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.
The zero marks the boundary, not the sign for the whole interval. The sign for each interval comes from a test point inside that interval.
Compact Written Example
For a small written version of the same idea, suppose
\[f(x)=(x+2)(x-1)(x-4).\]The zeros are
\[x=-2, \quad x=1, \quad x=4.\]Those zeros split the number line into four intervals:
\[(-\infty,-2), \quad (-2,1), \quad (1,4), \quad (4,\infty).\]| Factor or product | $(-\infty,-2)$ | $(-2,1)$ | $(1,4)$ | $(4,\infty)$ |
|---|---|---|---|---|
| $x+2$ | $-$ | $+$ | $+$ | $+$ |
| $x-1$ | $-$ | $-$ | $+$ | $+$ |
| $x-4$ | $-$ | $-$ | $-$ | $+$ |
| $f(x)$ | $-$ | $+$ | $-$ | $+$ |
The final row is the only row that tells you whether the polynomial itself is above or below the $x$-axis.
So for this written example:
\[f(x)<0 \quad \text{on} \quad (-\infty,-2) \cup (1,4)\]and
\[f(x)>0 \quad \text{on} \quad (-2,1) \cup (4,\infty).\]Why The Factor Rows Matter
Each factor row answers a smaller question. For example, the factor $x-4$ is negative to the left of $4$ and positive to the right of $4$.
The product row combines all of those smaller signs. That is why the video spends time plugging test points into the binomials before reading the final sign chart.
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: