- Create and evaluate a sign chart
Create and Evaluate a Sign Chart
sign chart
A sign chart is a visual tool that helps us determine where polynomial expressions are positive, negative, or zero by testing values in different intervals created by the polynomial’s zeros.
A sign chart displays the sign (positive or negative) of a polynomial expression across different intervals on the number line. A polynomial can only change signs when it equals zero.
How to: create a sign chart
- Factor the polynomial completely
- Find all zeros (where factors equal zero)
- Mark these zeros on a number line to create intervals
- Test a value from each interval
- Record the sign of each factor and the overall polynomial
Create a sign chart for [latex]f(x) = (x + 2)(x - 1)(x - 4)[/latex]
[latex]\begin{align} \text{Step 1: } &\text{Expression is already factored: } (x + 2)(x - 1)(x - 4) \\ \text{Step 2: } &\text{Find zeros by setting each factor equal to zero:}\\ &x + 2 = 0 \Rightarrow x = -2\\ &x - 1 = 0 \Rightarrow x = 1\\&x - 4 = 0 \Rightarrow x = 4\\ \text{Step 3: } &\text{Zeros are: } x = -2, 1, 4 \end{align}[/latex]
Step 4: Create intervals and test values
The zeros divide the number line into four intervals:
[latex]\\ \begin{align} (-\infty, -2): &\quad \text{Test } x = -3\\ (-2, 1): &\quad \text{Test } x = 0\\ (1, 4): &\quad \text{Test } x = 2\\ (4, \infty): &\quad \text{Test } x = 5 \end{align}[/latex]
Step 5: Evaluate each factor at test points
| Interval | Test Value | [latex](x+2)[/latex] | [latex](x-1)[/latex] | [latex](x-4)[/latex] | [latex]f(x)[/latex] |
|---|---|---|---|---|---|
| [latex](-\infty, -2)[/latex] | [latex]x = -3[/latex] | [latex](-3+2)[/latex] → − | [latex](-3-1)[/latex] → − | [latex](-3-4)[/latex] → − | (−)(−)(−) = − |
| [latex](-2, 1)[/latex] | [latex]x = 0[/latex] | [latex](0+2)[/latex] → + | [latex](0-1)[/latex] → − | [latex](0-4)[/latex] → − | (+)(−)(−) = + |
| [latex](1, 4)[/latex] | [latex]x = 2[/latex] | [latex](2+2)[/latex] → + | [latex](2-1)[/latex] → + | [latex](2-4)[/latex] → − | (+)(+)(−) = − |
| [latex](4, \infty)[/latex] | [latex]x = 5[/latex] | [latex](5+2)[/latex] → + | [latex](5-1)[/latex] → + | [latex](5-4)[/latex] → + | (+)(+)(+) = + |