Exponents & Polynomials
Adding & Subtracting Polynomials
Combining like terms to add and subtract polynomials.
Prerequisites
- Negative & Zero Exponents
Combining Whole Polynomials
Definition — Polynomial
Before reading on, predict: does adding two polynomials together require any new technique beyond combining like terms, the skill already used throughout this course on smaller expressions?
It doesn't — a polynomial is just a longer expression, and adding two of them means combining every pair of like terms across both, exactly the same rule from simplifying a single expression. Subtracting a polynomial takes one extra idea: distribute the negative sign across every term of the polynomial being subtracted before combining.
Worked Example — Adding Two Polynomials
Worked Example — Subtracting Two Polynomials
Tip
Common Mistakes
Distributing the negative sign to only the first term of the polynomial being subtracted, leaving the remaining terms' signs unchanged.
The negative sign in a subtraction applies to every single term of the second polynomial, not just the first — flip every sign before combining like terms.
Combining unlike-degree terms, such as adding a x² term directly to an x term.
Only terms with the exact same variable raised to the exact same power combine — an x² term and an x term stay as two separate terms in the final answer.
Key Takeaways
- Adding polynomials combines like terms across both, the same rule as combining like terms in any expression.
- Subtracting a polynomial requires distributing the negative sign across every one of its terms first.
- A polynomial's degree is its highest exponent.
Summary
Adding and subtracting polynomials reuses combining like terms at a larger scale. The next lesson multiplies polynomials together, extending the distributive property to expressions with more than one term on each side.
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