5.5 Graphs of Polynomials, Zeros & Multiplicity
Key Takeaways
- Connect algebraic zeros, linear factors (x - r), and geometric x-intercepts (r, 0): a polynomial f(x) has a zero at x = r if and only if (x - r) is a factor of f(x).
- Analyze the Multiplicity of Zeros: roots with odd multiplicity (1, 3, 5) cross the x-axis (linear cross or cubic inflection), while roots with even multiplicity (2, 4, 6) bounce off / are tangent to the x-axis.
- Determine global end behavior using the Leading Term Test: governed entirely by degree n (even vs odd) and the sign of leading coefficient a_n.
- Apply the Turning Point Theorem: a polynomial of degree n has at most n - 1 local turning points (extrema) and at most n real roots.
- Reconstruct polynomial equations from graphs: write f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}... and solve for vertical stretch factor a using the y-intercept or any known point.
Graphs of Polynomials, Zeros & Multiplicity
On the Digital SAT, polynomial questions frequently integrate graphical representations with algebraic expressions. You must be able to move seamlessly between a polynomial's factored algebraic form and its visual features on the coordinate plane, including its zeros, multiplicities, turning points, and end behavior.
1. Zeros, Factors & Intercepts Correspondence
For any polynomial function $f(x)$ with real coefficients, the following mathematical statements are completely equivalent:
+-----------------------------------------------------------------------------+
| EQUIVALENCE THEOREM OF POLYNOMIAL ROOTS |
| |
| ALGEBRAIC ROOT: x = r is a solution to f(x) = 0 |
| ▲ |
| │ |
| FACTOR FORM: (x - r) is a linear factor of f(x) |
| ▲ |
| │ |
| COORDINATE PLANE: (r, 0) is an x-intercept of the graph y = f(x) |
| ▲ |
| │ |
| REMAINDER THEOREM: f(r) = 0 (Remainder is 0 when divided by x - r) |
+-----------------------------------------------------------------------------+
2. Multiplicity of Zeros & Axis Behavior
The multiplicity of a zero is the number of times its corresponding linear factor appears in the fully factored form of the polynomial:
The multiplicity $m$ determines the geometric behavior of the graph at the $x$-intercept $(r, 0)$:
| Multiplicity ($m$) | Parity | Graph Behavior at $(r, 0)$ | Visual Curve Profile |
|---|---|---|---|
| $m = 1$ | Odd | Crosses straight through $x$-axis | Linear pass-through (no flattening) |
| $m = 2$ | Even | Bounces / Tangent to $x$-axis | Parabolic vertex touch without crossing |
| $m = 3$ | Odd | Crosses with Inflection | Flattens horizontally, then crosses through |
| $m = 4, 6, \dots$ | Even | Bounces with Flattening | Wider U-shape touch without crossing |
+-----------------------------------------------------------------------------+
| ROOT MULTIPLICITY VISUAL PROFILES |
| |
| [ MULTIPLICITY 1: CROSS ] [ MULTIPLICITY 2: BOUNCE ] |
| \ | |
| \ | |
| ----------*----------> ---------*---------> |
| \ / \ |
| \ / \ |
| |
| [ MULTIPLICITY 3: INFLECTION ] |
| ___/ |
| / |
| ----------*----------> |
| / |
| ___/ |
+-----------------------------------------------------------------------------+
3. The Leading Term Test for End Behavior
As $x$ approaches positive infinity ($x \to \infty$) or negative infinity ($x \to -\infty$), the value of a polynomial $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0$ is dominated entirely by its leading term $a_n x^n$ (the term with the highest power of $x$).
| Degree ($n$) | Leading Coeff ($a_n$) | Left End ($x \to -\infty$) | Right End ($x \to \infty$) | Visual Sketch |
|---|---|---|---|---|
| Even | Positive ($a_n > 0$) | $f(x) \to \infty$ (Up) | $f(x) \to \infty$ (Up) | $\nwarrow \quad \nearrow$ (Both Ends UP) |
| Even | Negative ($a_n < 0$) | $f(x) \to -\infty$ (Down) | $f(x) \to -\infty$ (Down) | $\swarrow \quad \searrow$ (Both Ends DOWN) |
| Odd | Positive ($a_n > 0$) | $f(x) \to -\infty$ (Down) | $f(x) \to \infty$ (Up) | $\swarrow \quad \nearrow$ (Down Left, Up Right) |
| Odd | Negative ($a_n < 0$) | $f(x) \to \infty$ (Up) | $f(x) \to -\infty$ (Down) | $\nwarrow \quad \searrow$ (Up Left, Down Right) |
+-----------------------------------------------------------------------------+
| END BEHAVIOR DECISION MATRIX |
| |
| DEGREE EVEN DEGREE ODD |
| +---------------------+ +---------------------+ |
| LEADING COEFF | ^ ^ | | ^ | |
| POSITIVE (> 0) | | | | | | | |
| | \_____/ | | \______/ | |
| | | | | | |
| | (Both Ends UP) | | (Down Left, Up Right| |
| +---------------------+ +---------------------+ |
| +---------------------+ +---------------------+ |
| LEADING COEFF | | | ^ | |
| NEGATIVE (< 0) | /-----\ | | | | |
| | | | | | \______ | |
| | v v | | | | |
| | (Both Ends DOWN) | | (Up Left, Down Right| |
| +---------------------+ +---------------------+ |
+-----------------------------------------------------------------------------+
4. Turning Points & Degree Constraints
- Maximum Turning Points: A polynomial function of degree $n$ has at most $n - 1$ turning points (local maxima and minima).
- Minimum Degree: If a graphed polynomial has $T$ turning points, its degree $n$ must be at least $T + 1$.
- Odd Degree Polynomials: Must have at least one real zero ($x$-intercept) because its ends point in opposite directions.
- Even Degree Polynomials: May have $0, 1, 2, \dots, n$ real zeros.
5. Step-by-Step Worked SAT Exam Examples
Worked Example 1: Reconstructing an Equation from a Graph
Problem: A polynomial $f(x)$ is graphed in the $xy$-plane. It has $x$-intercepts at $(-2, 0)$, $(1, 0)$, and $(3, 0)$. At $x = -2$, the graph crosses the axis linearly. At $x = 1$, the graph is tangent to (bounces off) the axis. At $x = 3$, the graph crosses linearly. The $y$-intercept is $(0, -6)$. What is the equation of $f(x)$?
Step-by-Step Solution:
- Identify factors and multiplicities:
- Root at $x = -2$ (cross) $\implies (x + 2)^1$
- Root at $x = 1$ (bounce) $\implies (x - 1)^2$
- Root at $x = 3$ (cross) $\implies (x - 3)^1$
- Write the general factored form with leading coefficient $a$:
- Use the $y$-intercept $(0, -6)$ to solve for $a$:
- State the final equation:
- Verify degree and end behavior: Total degree is $1 + 2 + 1 = 4$ (even). Since $a = 1 > 0$, both ends point up as $x \to \pm\infty$.
Worked Example 2: Determining Real Zeros from Factored Expressions
Problem: How many distinct real zeros does the function $g(x) = (x^2 - 9)(x^2 + 4)(x - 3)$ possess?
Step-by-Step Solution:
- Factor each term completely over the real numbers:
- $(x^2 - 9) = (x - 3)(x + 3)$
- $(x^2 + 4)$ has no real zeros (it yields imaginary roots $x = \pm 2i$)
- $(x - 3) = (x - 3)$
- Combine all real factors:
- List the distinct real roots:
- $x = 3$ (with multiplicity 2)
- $x = -3$ (with multiplicity 1)
- Conclusion: $g(x)$ has exactly $2$ distinct real zeros (at $x = 3$ and $x = -3$).
6. Desmos Testing Strategies & Visual Zero Finding
[!TIP] Counting Real Zeros and Multiplicities in Desmos: When asked for the number of distinct real zeros or to match a graph:
- Type the equation directly into Desmos:
y = (x^2 - 9)(x^2 - 6x + 9).- Desmos will highlight every $x$-intercept with a clickable gray dot.
- Count the dots to find the number of distinct real zeros.
- Inspect each zero visually:
- A straight crossing indicates multiplicity 1.
- A parabolic bounce indicates multiplicity 2.
- A flattened inflection crossing indicates multiplicity 3.
[!NOTE] Locating Local Extrema / Turning Points: Desmos automatically places clickable gray dots at every local maximum and local minimum. Click these points to read exact vertex/turning point coordinates.
The graph of a polynomial function $f(x)$ has $x$-intercepts at $(-4, 0)$, $(1, 0)$, and $(3, 0)$. The graph crosses the $x$-axis at $(-4, 0)$ and $(3, 0)$, but is tangent to (bounces off) the $x$-axis at $(1, 0)$. As $x \to \infty$, $f(x) \to \infty$, and as $x \to -\infty$, $f(x) \to \infty$. Which function could represent $f(x)$?
A polynomial function $g(x) = a x^5 + b x^4 + c x^3 + d x^2 + e x + f$ has a negative leading coefficient ($a < 0$). Which description correctly characterizes the end behavior of the graph of $g(x)$?
A cubic polynomial $P(x)$ has zeros at $x = -2$, $x = 1$, and $x = 4$, and its $y$-intercept is $(0, 24)$. What is the leading coefficient $a$ of $P(x)$, and what is the value of $P(2)$?