SAT — mathHard4 min read

Polynomial Functions

Analyse polynomial functions for the Digital SAT. Understand end behaviour, zeros, multiplicity, and the factor theorem.

Tutor AI Team2026-02-25T01:25:39.011Z

Polynomial functions go beyond quadratics to include cubic, quartic, and higher-degree polynomials. The Digital SAT tests your understanding of zeros (roots), end behaviour, the factor theorem, and how to interpret polynomial graphs. These are among the most challenging Advanced Math topics.

Core Concepts

Zeros/Roots

The zeros of $f(x)$ are the $x$ -values where $f(x) = 0$ — the x-intercepts of the graph.

If $f(x) = (x - 2)(x + 3)(x - 5)$ , the zeros are $x = 2, -3, 5$ .

The Factor Theorem

If $f(a) = 0$ , then $(x - a)$ is a factor of $f(x)$ . Conversely, if $(x - a)$ is a factor, then $f(a) = 0$ .

Example: If $f(x) = x^3 - 6x^2 + 11x - 6$ and $f(1) = 0$ , then $(x - 1)$ is a factor.

Multiplicity

Single root ( $x - a$ ): graph crosses the x-axis at $x = a$ .
Double root $(x - a)^2$ : graph touches and bounces at $x = a$ .
Triple root $(x - a)^3$ : graph flattens and crosses at $x = a$ .

End Behaviour

Determined by the leading term $ax^n$ :

Degree	Leading Coefficient	Left End	Right End
Even	Positive	↑	↑
Even	Negative	↓	↓
Odd	Positive	↓	↑
Odd	Negative	↑	↓

Number of Turning Points

A degree- $n$ polynomial has at most $n - 1$ turning points.

Degree and Number of Zeros

A degree- $n$ polynomial has at most $n$ real zeros (counting multiplicity).

Strategy Tips

Tip 1: Factor Theorem for Finding Zeros

If the SAT tells you $f(2) = 0$ , then $(x - 2)$ is a factor. Use polynomial division to find other factors.

Tip 2: End Behaviour from Leading Term

Only the highest-power term matters for end behaviour.

Tip 3: Match Graphs to Equations

Count x-intercepts and note end behaviour to match a graph with its equation.

Tip 4: Use Desmos

Type the polynomial into Desmos to see its graph, zeros, and turning points.

Tip 5: Multiplicity from Graphs

If the graph bounces at an x-intercept, it's a double root.

Worked Example: Example 1

Problem

Find the zeros of $f(x) = x^3 - 4x^2 - x + 4$ .

Test $x = 1$ : $1 - 4 - 1 + 4 = 0$ ✓. So $(x-1)$ is a factor.

Divide: $f(x) = (x-1)(x^2-3x-4) = (x-1)(x-4)(x+1)$ .

Zeros: $x = 1, 4, -1$ .

Solution

Worked Example: Example 2

Problem

What is the end behaviour of $f(x) = -2x^4 + 3x^2 - 1$ ?

Leading term: $-2x^4$ (even degree, negative). Both ends go down: $f(x) \to -\infty$ as $x \to \pm\infty$ .

Solution

Worked Example: Example 3

Problem

A graph crosses the x-axis at $x = -2$ and $x = 3$ , and touches (bounces) at $x = 1$ . Write a possible equation.

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

Solution

Worked Example: SAT-Style

Problem

If $f(x) = x^3 + 2x^2 - 5x - 6$ and $f(-1) = 0$ , which is a factor of $f(x)$ ?

Since $f(-1) = 0$ , $(x + 1)$ is a factor.

Solution

Worked Example: Example 5

Problem

A polynomial of degree 4 with positive leading coefficient. Describe the end behaviour.

Both ends point up: as $x \to -\infty$ , $f(x) \to +\infty$ , and as $x \to +\infty$ , $f(x) \to +\infty$ .

Solution

Practice Problems

Problem 1

Find all zeros of $f(x) = (x-1)(x+2)(x-4)$ .

Problem 2

What is the end behaviour of $f(x) = 5x^3 - x$ ?

Problem 3

A polynomial graph touches the x-axis at $x = 3$ and crosses at $x = -1$ . What is the minimum possible degree?

Problem 4

If $f(2) = 0$ for $f(x) = x^3 - 7x + 6$ , find all zeros.

Problem 5

How many turning points can a degree-5 polynomial have at most?

Want to check your answers and get step-by-step solutions?

Common Mistakes

Confusing the factor theorem direction. $f(a) = 0$ means $(x-a)$ is a factor, not $(x+a)$ .
Wrong end behaviour for negative leading coefficients. Negative flips the direction.
Forgetting multiplicity. A double root bounces, not crosses.
Thinking degree = number of real zeros. A degree-4 polynomial might have only 2 or 0 real zeros.

Key Takeaways

✓
Zeros are where $f(x) = 0$ — the x-intercepts.
✓
Factor theorem: $f(a) = 0$ ↔ $(x-a)$ is a factor.
✓
End behaviour depends on the degree (even/odd) and leading coefficient sign.
✓
Multiplicity: single root → crosses, double root → bounces.
✓
Max $n-1$ turning points for a degree- $n$ polynomial.

Tags:

#sat#math#advanced-math#polynomials#functions

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Core Concepts

Zeros/Roots

The Factor Theorem

Multiplicity

End Behaviour

Number of Turning Points

Degree and Number of Zeros

Strategy Tips

Tip 1: Factor Theorem for Finding Zeros

Tip 2: End Behaviour from Leading Term

Tip 3: Match Graphs to Equations

Tip 4: Use Desmos

Tip 5: Multiplicity from Graphs

Worked Example: Example 1

Worked Example: Example 2

Worked Example: Example 3

Worked Example: SAT-Style

Worked Example: Example 5

Practice Problems

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Common Mistakes

Key Takeaways

Related Topics

Parallel Structure

Verb Tense and Consistency

Modifier Placement and Dangling Modifiers

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