Zeros of Polynomials
to Construct Graphs
Learn to identify zeros from factored polynomials and use them to sketch accurate graphs showing intercepts, end behavior, and multiplicity.
What Is a Zero of a Polynomial?
A zero (or root) of a polynomial \( f(x) \) is a value \( x = c \) where \( f(c) = 0 \). Graphically, zeros are the x-intercepts: the points where the curve crosses or touches the x-axis.
Seeing Zeros on a Graph
Consider \( f(x) = (x + 2)(x - 1)(x - 3) \). It has three zeros: \( x = -2 \), \( x = 1 \), and \( x = 3 \). Each zero corresponds to an x-intercept.
Why Do Zeros Matter?
Zeros are the skeleton of a polynomial graph. Once you know where the curve touches the x-axis, you can determine the sign of \( f(x) \) in each interval, sketch end behavior, and draw a surprisingly accurate graph: no calculator needed.
The Factor-Zero Theorem
This works in reverse too. If \( x = c \) is a zero of \( f(x) \), then \( (x - c) \) is a factor. The theorem gives us a direct bridge between algebraic factors and graphical intercepts.
Understanding Multiplicity
When a factor \( (x - c) \) appears more than once, the zero \( x = c \) has a multiplicity greater than 1. Multiplicity controls how the graph behaves at that intercept.
End Behavior of Polynomials
End behavior describes what happens to \( f(x) \) as \( x \to +\infty \) and \( x \to -\infty \). It depends on the leading term \( a_n x^n \): the degree \( n \) and the sign of \( a_n \).
| \( a_n > 0 \) | \( a_n < 0 \) | |
|---|---|---|
| Even degree | Both ends rise: \( \uparrow \uparrow \) | Both ends fall: \( \downarrow \downarrow \) |
| Odd degree | Falls left, rises right: \( \downarrow \uparrow \) | Rises left, falls right: \( \uparrow \downarrow \) |
Zeros vs. Roots vs. x-intercepts
These three terms refer to the same concept from different perspectives. Recognizing when to use each helps you read problems correctly.
| Term | Perspective | Form |
|---|---|---|
| Zero | Function language | \( f(c) = 0 \) |
| Root | Equation language | \( c \) solves \( f(x) = 0 \) |
| x-intercept | Graph language | Point \( (c, 0) \) |
How to Graph from Zeros
Example 1: \( f(x) = x^2 - 4 \)
Example 2: \( g(x) = -x(x - 3)^2 \)
Example 3: \( h(x) = (x+1)(x-1)(x-4) \)
Zeros: \( x = -1 \), \( x = 1 \), \( x = 4 \): all multiplicity 1 (all cross). Degree 3, positive lead \( \to \) falls left, rises right (\( \downarrow \uparrow \)). The y-intercept is \( h(0) = (1)(-1)(-4) = 4 \).
Sign Analysis Between Zeros
Zeros split the x-axis into intervals. In each interval, \( f(x) \) is entirely positive or entirely negative (it can't change sign without crossing zero). Pick a test point in each interval to determine the sign.
Quartic Example: \( p(x) = x^2(x-2)(x+3) \)
Zeros: \( x = 0 \) (mult. 2, bounces), \( x = 2 \) (mult. 1, crosses), \( x = -3 \) (mult. 1, crosses). Degree 4, positive lead \( \to \) both ends rise (\( \uparrow \uparrow \)).
Common Mistakes to Avoid
Quick Reference Summary
| Feature | What to Check | Graph Effect |
|---|---|---|
| Zeros | Set each factor to 0 | x-intercepts on graph |
| Multiplicity (odd) | Exponent is 1, 3, 5, ... | Crosses x-axis |
| Multiplicity (even) | Exponent is 2, 4, 6, ... | Bounces off x-axis |
| End behavior | Degree + lead coefficient | Direction of "tails" |
| y-intercept | Evaluate \( f(0) \) | Point \( (0, f(0)) \) |
Question 1 of 5
What are the zeros of \( f(x) = (x - 5)(x + 2)(x - 1) \)?
Question 2 of 5
If \( g(x) = (x + 1)^2(x - 3) \), what happens at \( x = -1 \)?
Question 3 of 5
What is the end behavior of \( f(x) = -2x^3 + 5x^2 - x + 7 \)?
Question 4 of 5
How many times does the graph of \( h(x) = x(x-2)^2(x+4) \) cross the x-axis?
Question 5 of 5
What is the y-intercept of \( f(x) = 2(x - 1)(x + 3)(x - 2) \)?