How to Simplify Polynomials? (+FREE Worksheet!)

Simplifying polynomials is one of the most fundamental skills in Algebra 1. By identifying and combining like terms, you can reduce any messy polynomial expression to its simplest, most useful form. This guide walks through the concept step by step, with worked examples, two video lessons, and practice problems so you can build real confidence.

What Is Simplifying a Polynomial?

A polynomial is an expression made up of terms that contain variables raised to whole-number powers and real-number coefficients — for example, \(\color{blue}{3x^{2} + 5x – 2}\). Simplifying a polynomial means rewriting it in its most compact form by combining any like terms: terms that share the exact same variable(s) raised to the exact same power(s).

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How to Simplify Polynomials

1. Identify Like Terms

Like terms have identical variable parts. In \(\color{blue}{3x^{2} + 5x – 2 + 4x^{2} – 3x + 7}\), the like term pairs are \(\color{blue}{3x^{2}}\) and \(\color{blue}{4x^{2}}\); \(\color{blue}{5x}\) and \(\color{blue}{-3x}\); and the constants \(\color{blue}{-2}\) and \(\color{blue}{7}\).

2. Add or Subtract the Coefficients

Keep the variable part unchanged and add (or subtract) only the numerical coefficients of like terms.

• \(\color{blue}{3x^{2} + 4x^{2} = 7x^{2}}\)
• \(\color{blue}{5x + (-3x) = 2x}\)
• \(\color{blue}{-2 + 7 = 5}\)

Result: \(\color{blue}{7x^{2} + 2x + 5}\)

3. Write Terms in Descending Order

Standard form arranges terms from the highest power to the lowest. Write the simplified polynomial in descending order of degree for clean, readable results.

Step-by-Step Summary

1. Identify every term and its variable part.
2. Group like terms together (same variable, same exponent).
3. Add or subtract the coefficients within each group.
4. Write the result in standard (descending) form.

Watch: Simplifying Polynomials — Concept Lesson

This Math Antics video explains like terms and shows how to combine them clearly:

Simplifying Polynomials — Worked Examples

Example 1: Simplify \(\color{blue}{3x^{2} + 5x – 2 + 4x^{2} – 3x + 7}\).

Group like terms: \(\color{blue}{(3x^{2} + 4x^{2}) + (5x – 3x) + (-2 + 7)}\)
Add coefficients: \(\color{blue}{7x^{2} + 2x + 5}\)

Example 2: Simplify \(\color{blue}{2x^{3} – x^{2} + 4x – 6 – x^{3} + 3x^{2} – 2x + 1}\).

Group: \(\color{blue}{(2x^{3} – x^{3}) + (-x^{2} + 3x^{2}) + (4x – 2x) + (-6 + 1)}\)
Combine: \(\color{blue}{x^{3} + 2x^{2} + 2x – 5}\)

Example 3: Simplify \(\color{blue}{4a^{2}b – 3\text{ ab }^{2} + 2\text{ ab } – a^{2}b + 5\text{ ab }^{2} – \text{ ab }}\).

Group: \(\color{blue}{(4a^{2}b – a^{2}b) + (-3\text{ ab }^{2} + 5\text{ ab }^{2}) + (2\text{ ab } – \text{ ab })}\)
Combine: \(\color{blue}{3a^{2}b + 2\text{ ab }^{2} + \text{ ab }}\)

Example 4: Simplify \(\color{blue}{(6x^{2} + x – 8) – (2x^{2} – 3x + 2)}\).

Distribute the negative: \(\color{blue}{6x^{2} + x – 8 – 2x^{2} + 3x – 2}\)
Group: \(\color{blue}{(6x^{2} – 2x^{2}) + (x + 3x) + (-8 – 2)}\)
Combine: \(\color{blue}{4x^{2} + 4x – 10}\)

More Examples: Step-by-Step Video Review

Want a second walkthrough? Khan Academy works through additional simplification problems:

Exercises for Simplifying Polynomials

Simplify each expression by combining like terms.

1. \(\color{blue}{5x^{2} – 2x + 3 + x^{2} + 4x – 1}\)
2. \(\color{blue}{3y^{3} – y + 2 – 2y^{3} + 4y – 5}\)
3. \(\color{blue}{(2m^{2} – 3m + 1) + (m^{2} + 5m – 4)}\)
4. \(\color{blue}{(6x^{2} + x – 8) – (2x^{2} – 3x + 2)}\)
5. \(\color{blue}{4\text{ ab } – 2a + 3b – \text{ ab } + 5a – b}\)

Answers

1. \(\color{blue}{6x^{2} + 2x + 2}\)
2. \(\color{blue}{y^{3} + 3y – 3}\)
3. \(\color{blue}{3m^{2} + 2m – 3}\)
4. \(\color{blue}{4x^{2} + 4x – 10}\)
5. \(\color{blue}{3\text{ ab } + 3a + 2b}\)

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Want More Practice?

We haven’t published a worksheet built specifically for Simplifying Polynomials just yet. In the meantime, the free worksheets below cover closely related skills and concepts. If you’d like extra practice, download any that look helpful, complete the problems, and check your work — they’re a great way to reinforce what you learned on this page and strengthen the foundations this topic builds on:

• Download Adding and Subtracting Polynomials Worksheet
• Download Multiplying Polynomials Worksheet
• Download Greatest Common Factor and GCF Factoring Worksheet

Frequently Asked Questions

What does it mean to simplify a polynomial?

Simplifying a polynomial means combining all like terms so the expression has as few terms as possible and is written in standard (descending) form. For example, \(\color{blue}{2x + 3x}\) simplifies to \(\color{blue}{5x}\).

Can you combine x² terms with x terms?

No. Like terms must share the same variable and the same exponent. \(\color{blue}{3x^{2}}\) and \(\color{blue}{5x}\) are not like terms and cannot be combined.

What is standard form for a polynomial?

Standard form lists terms in order from the highest exponent to the lowest. For example, \(\color{blue}{4x^{3} – 2x^{2} + x – 7}\) is in standard form.

Related Topics

• Adding and Subtracting Polynomials
• Multiplying Monomials
• Multiplying Binomials
• Factoring Trinomials
• Factoring Polynomials