What Is a Polynomial?

Polynomial

A polynomial is an algebraic expression formed by adding or subtracting terms, where each term is a constant multiplied by a variable raised to a non-negative whole-number power. Example: \(4x^{3} - 2x + 7\).

Parts of a Polynomial

Polynomial Components

A polynomial is built from separate terms. Each term combines a coefficient, a variable, and a whole-number exponent.

Key Characteristics:

  • Terms: \(5x^{2},\,-3x,\,8\) — parts added or subtracted.
  • Coefficients: \(5,\,-3,\,8\) — numbers in front of the variable.
  • Variable: \(x\) — symbol that can change value.
  • Exponents: \(2\) and \(1\) — whole numbers showing repeated multiplication.

Example:

In \(5x^{2}-3x+8\), you can now point out every term, coefficient, variable, and exponent.

Finding the Degree

1

Check Exponents

Write down the exponent of each term.

2

Find the Highest

Pick the largest exponent you see.

3

State the Degree

That highest exponent is the polynomial's degree.

Pro Tip:

Example: For \(7x^3 + 4x^2 - x\), the highest exponent is 3, so the degree is 3.

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Common Types

Based on degree

Linear (1st degree)

Highest power is 1; e.g., \(2x + 5\).

Quadratic (2nd degree)

Highest power is 2; e.g., \(x^{2} - 4x + 3\).

Cubic (3rd degree)

Highest power is 3; e.g., \(-x^{3} + 6\).

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Is it a Polynomial?

Drag each expression to the right bucket to practise classification.

Draggable Items

\(3x^2 + 1\)
\(\frac{2}{x}\)
\(5y - 4\)
\(\sqrt{x} + 7\)

Drop Zones

Polynomial ✔️

Not a Polynomial ❌

Tip:

A polynomial uses only +, −, × with variables raised to whole-number powers. Roots or variables in denominators break the rule.

Key Takeaways

A polynomial is a sum of terms with non-negative integer exponents.

Each term has a coefficient, a variable, and an exponent.

The degree of a polynomial is its highest exponent.

Common types: linear (degree 1), quadratic (2), cubic (3).

Variables cannot be in denominators or under roots.

Thank You!

We hope you found this lesson informative and engaging.