Polynomial Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Polynomial Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Functions made by adding terms of the form ax^n (where n is a non-negative integer).

Sums of power terms with whole-number exponents. The building blocks of functions.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The degree (highest power) determines the function's basic shape.

Common stuck point: A degree n polynomial has at most n roots and n-1 turning points.

Sense of Study hint: Identify the degree and leading coefficient first. Then check end behavior: does it go up-up, down-down, or up-down?

Worked Examples

Example 1

easy
Find the degree and leading coefficient of p(x) = -3x^4 + 7x^2 - x + 5.

Solution

  1. 1
    Write the polynomial in descending powers and identify the highest exponent present.
  2. 2
    The highest power is x^4, so the degree is 4.
  3. 3
    The coefficient attached to the leading term -3x^4 is -3, so the leading coefficient is -3.

Answer

\text{Degree } 4, \quad \text{leading coefficient } -3
The degree determines the end behavior and maximum number of roots. A negative leading coefficient with even degree means the graph falls on both ends.

Example 2

medium
Find all zeros of p(x) = x^3 - 4x^2 + x + 6.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
Perform polynomial long division: \frac{2x^3 + 3x^2 - 5x + 1}{x + 2}.

Example 2

hard
Sketch the end behavior and find all real zeros of f(x) = -2x^3 + 6x^2 + 8x. State the multiplicity of each zero.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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