Adding polynomials

Apples with apples, pears with pears

An expression with more terms has a name: a polynomial. 3x + 2 is a binomial, 2x² + 5x − 7 a trinomial. Nothing scary — just expressions made of several terms.

Today’s question: how do you add two polynomials? Say (3x + 2) + (5x + 4)?

Every fruit seller knows the answer. 3 apples and 2 pears plus 5 apples and 4 pears = 8 apples and 6 pears. Apples with apples, pears with pears. Never apples with pears.

In maths this is called collecting like terms: 3x + 5x = 8x, because both terms have the same letter. But 3x + 4 cannot be collected — x and a number are different “fruit”.

The only trick today is a minus in front of brackets. It flips the signs of everything inside. Anyone who can do that has adding and subtracting polynomials in their pocket.

Filip and Tereza are counting material for two cabins from a building set. First cabin: 4 long rods and 6 short ones. Second: 3 long and 8 short. Filip writes on paper: 4d + 6k and under it 3d + 8k. “Altogether?” Tereza adds by type: “Long: 4 + 3 = 7. Short: 6 + 8 = 14. So 7d + 14k.” Filip tries to add 7d + 14k into “21dk” — and Tereza stops him: “A long rod plus a short rod is not a new rod. Different types are not added, they are just written next to each other.”

How to add and subtract polynomials

Adding:

1. Remove the brackets. With a plus in front of the brackets nothing changes: (3x + 2) + (5x + 4) = 3x + 2 + 5x + 4.
2. Find like terms. It helps to underline them: x-terms with a straight line, numbers with a wavy one.
3. Collect: 3x + 5x = 8x and 2 + 4 = 6. Result: 8x + 6.

Subtracting — watch, a minus in front of the brackets:

(7x + 5) − (2x + 3): the minus flips the sign of every term in the brackets. You get 7x + 5 − 2x − 3. Collect: 5x + 2.

Why does it flip? Subtracting a pack means subtracting everything in it. You subtract 2x and you subtract 3 as well.

With powers: (2x² + 3x) + (x² − 5x) = 2x² + x² + 3x − 5x = 3x² − 2x. Terms with x² together, terms with x together. x² and x are never collected.

At the end check the number of types: however many different letter parts there were at the start, that is the most there may be at the end.

Example 1: (4a + 7) + (3a − 2)

Plus in front of the brackets — the brackets just disappear:

4a + 7 + 3a − 2.

I underline types: a-terms 4a and 3a, numbers +7 and −2.

I collect: 4a + 3a = 7a and 7 − 2 = 5.

Result: 7a + 5.

Check by substituting: for a = 1 the original writing gives (4 + 7) + (3 − 2) = 12 and the result 7 + 5 = 12. It fits.

Example 2: (6x + 4) − (2x + 9)

Minus in front of the brackets — I flip the signs of the whole second pair of brackets:

6x + 4 − 2x − 9.

I collect: 6x − 2x = 4x and 4 − 9 = −5.

Result: 4x − 5.

The most common mistake would be 6x + 4 − 2x + 9: the minus is used only on the first term and the nine stays with a plus. So: a minus in front of brackets = flip everything inside.

Example 3: (3x² + 2x − 1) + (x² − 2x + 5)

Three types of terms: x², x and numbers.

I remove the brackets: 3x² + 2x − 1 + x² − 2x + 5.

x²-terms: 3x² + x² = 4x². x-terms: 2x − 2x = 0 — they cancel completely and disappear. Numbers: −1 + 5 = 4.

Result: 4x² + 4.

That some type cancels to zero is normal. Zero is not written in the result.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Practise adding and subtracting negative numbers — that is exactly what you do with coefficients when collecting.

Practise the questions

On paper

Underline types of terms. The hand gets used to it and then the eyes will see a mistake on their own.

1. Collect: 5x + 3x, 9a − 4a, 2b + 7b − b. Write the mid-step with underlined types too.

2. Calculate (8x + 3) + (2x − 7) and check by substituting x = 1 into the question and the result.

3. Calculate (5y + 6) − (3y − 4). Watch both signs after the minus.

4. Calculate (2x² + x + 3) + (x² − x + 1) and write which type of terms cancelled.

Now you 💪

1. I collect only terms with the same letter part.
2. A minus in front of brackets flips the signs of all terms inside.
3. I can check the result by substituting a number.

What to take from this lesson

• A polynomial is an expression with more terms: 2x² + 5x − 7.
• You can only collect like terms: 3x + 5x = 8x, but 3x + 4 stays.
• x² and x are different types — they are not collected.
• A minus in front of brackets flips the sign of every term inside.
• A check by substituting spots a bad collection in a few seconds.

© 2026 Ing. Martin Polak / AlgoRhino · Content usage terms