Adding and Subtracting Polynomials Calculator

RESULT

—

Choose + or −, enter two polynomials, and press =.

In columns

Operation

First polynomial

Second polynomial

Every power gets its own column, so a missing term shows as a gap rather than quietly shifting the others along. Subtracting flips the sign of every term in the second polynomial.

Examples — tap one and it solves

How the answer is built, step by step

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No one thinks this topic is hard. That is exactly why marks disappear in it.

Adding and subtracting polynomials involves no method worth the name — you line the terms up and combine the ones that match. And yet a subtraction question will lose more marks in a class than a factorisation will, because the mistake is not in the thinking. It is a sign, written down in half a second, on a line nobody checks afterwards. This adding and subtracting polynomials calculator puts that line on screen deliberately.

Adding Polynomials: Only Like Terms Combine

Terms combine when they carry the same variable to the same power, and not otherwise. 5x³ and −2x³ are like terms and give 3x³. 5x³ and 5x² are not, and no amount of staring will make them one term.

Notice what does not change: the power. You add the coefficients and copy the power across. In a hurry it is oddly easy to write x² + x² = x⁴, which is multiplication's rule wandering into the wrong question.

Set It Out in Columns

Give every power its own column, and the pairing does itself:

        x²      x     1
first   3x²    2x   −1
second    x²  −4x    5
        ——————————————
total   4x²  −2x    4

This looks like fussiness until one polynomial is missing a term. x³ + 1 has no x² and no x; if you write the two side by side without columns, it is genuinely easy to add the wrong pair. With columns the gap is a blank space you can see, and the term from the other polynomial simply comes down.

Subtraction: Every Sign, Not Just the First

Here is the line that costs marks:

−(x² − 4x + 5) written as −x² − 4x + 5	✗ only the first sign changed
−(x² − 4x + 5) written as −x² + 4x − 5	✓ all three changed

The minus belongs to the bracket, so it reaches every term inside. That is the entire difficulty of subtracting polynomials, and it is why the bracket is worth writing even when it feels unnecessary — the bracket is the reminder.

The calculator prints the second polynomial twice: as you typed it, and again in red after the minus has been through it. Two rows, and the comparison is immediate.

The Degree Can Go Down

(x² + x) − (x² + 1) = x − 1. Two quadratics went in and something linear came out, because the x² terms cancelled.

This surprises people, and it is worth being comfortable with: adding or subtracting can lower the degree, and can never raise it. Nothing in either operation produces a bigger power than the ones you started with. Multiplication raises the degree; this does not.

Take it further and 5x² − 5x² leaves nothing at all — the answer is zero, not 0x². A coefficient of zero means the term is gone from the expression, not sitting there invisibly.

How to Use This Adding and Subtracting Polynomials Calculator

It works as an adding polynomials calculator and a subtracting one from the same screen: choose add or subtract, put a polynomial in each box, and press =. Terms can be in any order and brackets are expanded first. If you switch between + and − with an answer already showing, it re-answers straight away without you retyping anything.

The band gives the result and tells you the degree, how many gaps the columns had, and whether anything cancelled. Tap the button below and the columns open: the two polynomials, the flipped row in red if you are subtracting, and the totals underneath. The steps run through the same thing more slowly, ending with a note on the degree.

Once you are comfortable with this, multiplying polynomials is the natural next step — and it is where the degree starts going up instead.

Worked Examples

Question	Answer	Watch for
(3x² + 2x − 1) + (x² − 4x + 5)	4x² − 2x + 4	the plain case
(3x² + 2x − 1) − (x² − 4x + 5)	2x² + 6x − 6	all three signs flip
(x³ + 1) + (x² + x)	x³ + x² + x + 1	four gaps in the columns
(x² + x) − (x² + 1)	x − 1	the degree drops to 1
5x² − 5x²	0	nothing left at all

Rows one and two are the same two polynomials. The only difference is the sign in the middle, and the answers share nothing — which is a fair measure of how much that one line matters.

Frequently Asked Questions

How do you add polynomials?

Add the coefficients of terms that carry the same power, and leave the power itself alone. 3x² and x² add to 4x²; 3x² and 2x do not combine at all. Setting the two polynomials out in columns by power makes it obvious which terms pair up.

How do you subtract polynomials?

Change the sign of every term in the second polynomial, then add as usual. Subtracting x² − 4x + 5 means adding −x² + 4x − 5. Changing only the first sign and copying the rest is the single commonest mistake in this topic.

What are like terms?

Terms carrying the same variable raised to the same power. 5x³ and −2x³ are like terms and combine to 3x³. 5x³ and 5x² are not, and never combine however similar they look.

Can the degree change when you add or subtract polynomials?

It can drop, but never rise. If the leading terms cancel — as in (x² + x) − (x² + 1), which gives x − 1 — the answer has a lower degree than either polynomial you started with. It can never go higher, because there is nothing to produce a bigger power.

What happens if a term is missing from one polynomial?

Nothing, as long as you keep the columns lined up. A missing term is a coefficient of zero, so the term from the other polynomial comes straight down unchanged. Trouble only starts when the gap is closed up and terms end up under the wrong power.

Do you need brackets when subtracting polynomials?

Yes, and they are the whole point. The minus sign applies to the bracket, not just to the first term inside it, and writing the bracket is what reminds you of that. Drop the bracket too early and the signs go wrong.

Is this adding and subtracting polynomials calculator free?

Completely free, with the columns and every step shown, and no limit on how many you work through. No account, no download, and nothing held back behind a subscription.

Two habits cover almost everything here. Write the columns even when the polynomials look short enough to do in your head, and write the second polynomial out again with its signs changed before you add anything. Neither takes more than a few seconds, and between them they remove nearly every mark that goes missing in this topic.