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.
Do Operations with Polynomials: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Do Operations with Polynomials: pop-up practice
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: \(x^2 + x^2 = 2x^2\).
Related Topics
Continue Your Study
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