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 rewriting it with as few terms as possible by combining like terms, then putting the terms in standard form. The value of the expression does not change, only how it looks. For example, 6x² − 2x + 4 + x − 3x² simplifies to 3x² − x + 4, because the x² terms combine to 3x² and the x terms combine to −x.

How do you simplify polynomials step by step?

Step 1: remove parentheses, using the distributive property where needed. Step 2: group like terms, which have the same variable and exponent. Step 3: add or subtract their coefficients. Step 4: write the answer in standard form. For 2(x² + 3x) − 4x + 7, distribute to get 2x² + 6x − 4x + 7, then combine the x terms to get 2x² + 2x + 7.

How do you simplify polynomials with parentheses?

Distribute whatever is in front of each set of parentheses, paying close attention to minus signs, then combine like terms. A minus in front of parentheses changes the sign of every term inside. For example, 5y − (2y² − 3y + 1) becomes 5y − 2y² + 3y − 1, which simplifies to −2y² + 8y − 1. Forgetting to flip the sign of the last term is the most common slip.

How do you simplify polynomials into standard form?

After combining like terms, arrange the terms from the highest exponent to the lowest, with the constant last. The exponent of the first term is the polynomial’s degree, and its coefficient is the leading coefficient. For instance, 4 + 3x − x³ + 2x² reorders to −x³ + 2x² + 3x + 4, a third-degree polynomial with a leading coefficient of −1.

Is simplifying the same as factoring?

No, they go in nearly opposite directions. Simplifying combines like terms so an expression has fewer terms, while factoring rewrites an expression as a product of simpler pieces. For example, simplifying x² + 3x + 2x + 6 gives x² + 5x + 6, and factoring that result gives (x + 2)(x + 3). Both forms are equal; which one you want depends on the question.

Can you simplify polynomials on a calculator?

Some calculators can. A graphing tool like Desmos can check your work: graph the original and the simplified expression, and if the graphs match, the simplification is right. Calculators with a computer algebra system, such as the TI-Nspire CAS, can simplify directly. Most basic scientific calculators cannot, so practice by hand, especially for classes and tests that limit calculator use.

Related Topics

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