5.5 Graphs of Polynomials, Zeros & Multiplicity

Q: 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)?

f(x) = 0.5(x + 4)(x - 1)^2(x - 3). Map each x-intercept to its corresponding linear factor and multiplicity: an intercept at x = -4 that crosses corresponds to (x + 4)^1; an intercept at x = 1 that is tangent (bounces) corresponds to (x - 1)^2; and an intercept at x = 3 that crosses corresponds to (x - 3)^1. Combining these yields f(x) = a(x + 4)(x - 1)^2(x - 3), which has a total degree of 1 + 2 + 1 = 4 (even). Since both ends point upward (f(x) \to \infty as x \to \pm\infty), the leading coefficient must be positive (a = 0.5 > 0).

Q: 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)?

As x \to -\infty, g(x) \to \infty; and as x \to \infty, g(x) \to -\infty.. The polynomial has degree 5, which is an odd degree. For odd-degree polynomials with a negative leading coefficient (a < 0), the end behavior is dominated by the leading term a x^5. As x \to -\infty, x^5 becomes large and negative, so multiplying by negative a results in g(x) \to \infty. As x \to \infty, x^5 is large and positive, so multiplying by negative a results in g(x) \to -\infty. Thus, the graph points up to the left and down to the right.

Q: 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)?

a = 3, and P(2) = -24. Since P(x) is a cubic polynomial with zeros at -2, 1, 4, its factored form is P(x) = a(x + 2)(x - 1)(x - 4). Use the y-intercept (0, 24) to determine a: P(0) = a(0 + 2)(0 - 1)(0 - 4) = 24 \implies a(2)(-1)(-4) = 24 \implies 8a = 24 \implies a = 3. The complete function is P(x) = 3(x + 2)(x - 1)(x - 4). Now evaluate P(2): P(2) = 3(2 + 2)(2 - 1)(2 - 4) = 3(4)(1)(-2) = -24.

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.

Last updated: August 2026

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:

f(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2} \cdots (x - r_k)^{m_k}

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 $a$ : $f(x) = a(x + 2)(x - 1)^2(x - 3)$
Use the $y$ -intercept $(0, -6)$ to solve for $a$ : $f(0) = a(0 + 2)(0 - 1)^2(0 - 3) = -6$ $a(2)(1)(-3) = -6 \implies -6a = -6 \implies a = 1$
State the final equation: $f(x) = (x + 2)(x - 1)^2(x - 3)$
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: $g(x) = (x - 3)^2 (x + 3) (x^2 + 4)$
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.

Test Your Knowledge

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 $f(x)$ ?

$f(x) = 0.5(x + 4)(x - 1)^2(x - 3)$

$f(x) = -0.5(x + 4)^2(x - 1)(x - 3)$

$f(x) = 0.5(x - 4)(x + 1)^2(x + 3)$

$f(x) = 0.5(x + 4)^2(x - 1)^2(x - 3)^2$

Test Your Knowledge

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 $g(x)$ ?

As $x \to -\infty$ , $g(x) \to -\infty$ ; and as $x \to \infty$ , $g(x) \to \infty$ .

As $x \to -\infty$ , $g(x) \to \infty$ ; and as $x \to \infty$ , $g(x) \to \infty$ .

As $x \to -\infty$ , $g(x) \to -\infty$ ; and as $x \to \infty$ , $g(x) \to -\infty$ .

As $x \to -\infty$ , $g(x) \to \infty$ ; and as $x \to \infty$ , $g(x) \to -\infty$ .

Test Your Knowledge

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 $P(2)$ ?

$a = -3$ , and $P(2) = 24$

$a = 3$ , and $P(2) = -24$

$a = 2$ , and $P(2) = -16$

$a = 4$ , and $P(2) = -32$

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5.6 Function Notation, Composition & Transformations

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SAT Math

5.5 Graphs of Polynomials, Zeros & Multiplicity

Key Takeaways

Graphs of Polynomials, Zeros & Multiplicity

1. Zeros, Factors & Intercepts Correspondence

2. Multiplicity of Zeros & Axis Behavior

3. The Leading Term Test for End Behavior

4. Turning Points & Degree Constraints

5. Step-by-Step Worked SAT Exam Examples

Worked Example 1: Reconstructing an Equation from a Graph

Worked Example 2: Determining Real Zeros from Factored Expressions

6. Desmos Testing Strategies & Visual Zero Finding