Analyzing Polynomial Zeros and Intervals of Positivity/Negativity

Q: For a polynomial function, how do you determine the intervals where the function is positive or negative?

Background

Topic: Polynomial Functions – Zeros and Sign Analysis

This question is testing your ability to analyze a polynomial function by finding its zeros and determining on which intervals the function is positive or negative. This is a key skill in precalculus for understanding the behavior of polynomial graphs.

Key Terms and Formulas

• Zero of a function: A value of $x$ where $f(x) = 0$.

• Interval: A range of $x$-values between zeros (or extending to infinity).

• Test value: A value chosen within an interval to determine the sign of $f(x)$.

• Sign of $f(x)$: Whether $f(x)$ is positive or negative in that interval.

Step-by-Step Guidance

1. Find all the real zeros of the polynomial function $f(x)$. These are the $x$-values where $f(x) = 0$.

2. Write the intervals determined by these zeros. For example, if the zeros are $x_1$, $x_2$, and $x_3$, the intervals will be $(-\infty, x_1)$, $(x_1, x_2)$, $(x_2, x_3)$, and $(x_3, \infty)$.

3. Choose a test value $x$ within each interval. Substitute this value into $f(x)$ to determine whether $f(x)$ is positive or negative in that interval.

4. Record the sign of $f(x)$ for each interval in a table like the one shown above. This helps you visualize where the function is above or below the $x$-axis.

Try solving on your own before revealing the answer!

Final Answer:

After finding the zeros and testing values in each interval, you can fill out the table to show where $f(x)$ is positive or negative. For example, if the zeros are $-2$, $0$, and $1$, the intervals are $(-\infty, -2)$, $(-2, 0)$, $(0, 1)$, and $(1, \infty)$, and you would test values like $-3$, $-1$, $0.5$, and $2$ in each interval.

The sign of $f(x)$ in each interval tells you where the graph is above or below the $x$-axis.