How to Do Operations with Polynomials? (+FREE Worksheet!)
How to Perform Operations With Polynomials
Adding, subtracting, and multiplying polynomials all come down to two skills you already have: combining like terms and distributing. Line up the same powers, and the rest is careful bookkeeping. We’ll work through each operation with a solver, a worksheet maker, and flashcards a tap away.
Operations with polynomials — adding, subtracting, and multiplying them — look harder than they are. Every one comes down to two moves: combining like terms and distributing. Line up matching powers of \(x\), keep your signs honest, and the rest is bookkeeping.
What Are Polynomial Operations?
A polynomial is a sum of terms like \(3x^2\), \(-5x\), and \(4\). “Like terms” share the same variable power — \(3x^2\) and \(x^2\) are like; \(3x^2\) and \(3x\) are not. Adding and subtracting just combine like terms; multiplying distributes every term of one polynomial across the other.
One habit pays off everywhere: write your answer in standard form — terms in descending order of power, highest first, like \(4x^2 – 3x + 3\). That’s how every answer key expects to see it.
The three operations at a glance:
- Add: combine like terms.
- Subtract: distribute the minus sign first, then combine like terms.
- Multiply: distribute every term, then combine like terms.
How Each Operation Works
Stack like terms
Combine matching powers.
\(=\) \(4x^2-3x+3\)
Flip the signs first
Distribute the minus to every term.
\(=\) \(2x^2+7x-5\)
Distribute, then combine
Every term times every term.
\(=\) \(x^2-2x-15\)
Worked Examples
A. Adding
\((3x^2+2x-1)+(x^2-5x+4)\).
Combine like terms: \(3x^2+x^2=4x^2\), \(2x-5x=-3x\), \(-1+4=3\). Result: \(4x^2-3x+3\).
B. Subtracting
\((3x^2+2x-1)-(x^2-5x+4)\).
Add the opposite: \(3x^2+2x-1-x^2+5x-4\). Combine: \(2x^2+7x-5\). Notice every sign in the second group flipped.
C. Monomial × polynomial
\(2x(3x^2-4x+5)\).
Distribute \(2x\): \(6x^3-8x^2+10x\). Each term gets multiplied. Result: \(6x^3-8x^2+10x\).
D. Two binomials (FOIL)
\((x+3)(x-5)\).
First/Outer/Inner/Last: \(x^2-5x+3x-15 = x^2-2x-15\). Result: \(x^2-2x-15\).
E. Binomial × trinomial
\((2x-1)(x^2+3x-2)\).
Distribute each term of \((2x-1)\): \(2x^3+6x^2-4x \;-\; x^2-3x+2\). Combine like terms: \(2x^3+5x^2-7x+2\). Same method, more terms — multiply every pair and don’t skip one.
F. Difference of squares
\((x+4)(x-4)\).
The outer and inner terms cancel: \(x^2-4x+4x-16 = x^2-16\). Result: \(x^2-16\). \((a+b)(a-b)=a^2-b^2\) is a pattern worth memorizing.
Slip-Ups That Cost Easy Points
- Dropping the subtraction sign. \(-(x^2-5x+4)\) is \(-x^2+5x-4\) — every term flips, not just the first. This is the #1 polynomial error.
- Combining unlike terms. \(3x^2\) and \(2x\) do not combine. Only terms with the exact same power add together.
- Forgetting a term when multiplying. Every term of the first polynomial must hit every term of the second. Be systematic, especially beyond FOIL.
- Adding exponents when you shouldn’t. When multiplying, \(x^2 \cdot x^3 = x^5\) (add exponents); when adding like terms, \(x^2 + x^2 = 2x^2\) (the power stays). Don’t mix the rules.
- Squaring a binomial as just two squares. \((x-3)^2 = x^2-6x+9\), not \(x^2-9\) — write it as \((x-3)(x-3)\) and FOIL; the middle term is real.
Your Turn: Simplify
Perform each operation and combine like terms. Reveal to check.
- \((x^2+3x)+(2x^2-x)\)
- \((5x^2-2x+1)-(3x^2+x-4)\)
- \(3x(x^2-2)\)
- \((x+4)(x+2)\)
- \((x-3)(x-3)\)
- \((2x+1)(x-5)\)
Show answers
- \(\color{blue}{3x^2+2x}\)
- \(\color{blue}{2x^2-3x+5}\)
- \(\color{blue}{3x^3-6x}\)
- \(\color{blue}{x^2+6x+8}\)
- \(\color{blue}{x^2-6x+9}\)
- \(\color{blue}{2x^2-9x-5}\)
Make Your Own Polynomials Worksheet
Generate fresh polynomial-operation problems with a full answer key — print or save as a PDF.
Frequently Asked Questions
What are “like terms” in a polynomial?
Terms with the exact same variable raised to the exact same power — \(4x^2\) and \(-x^2\) are like terms, but \(4x^2\) and \(4x\) are not. Only like terms can be added or subtracted.
How do I subtract polynomials without sign errors?
Rewrite the subtraction as adding the opposite: change every sign in the second polynomial, then combine like terms. Distributing the minus to every term is the step people skip.
What is FOIL and when do I use it?
FOIL (First, Outer, Inner, Last) is a way to remember to multiply every pair when multiplying two binomials. For larger products, the idea is the same — distribute every term — but FOIL only names the four products of two binomials.
Do exponents add when I multiply terms?
Yes, for the same base: \(x^2 \cdot x^3 = x^5\). But when you add like terms the exponent stays the same: \(x^2 + x^2 = 2x^2\).
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