Adding and Subtracting Polynomials Calculator
Our free algebra calculator solves adding subtracting polynomials problems. Get worked examples, visual aids, and downloadable results.
Formula
(a1x^2 + b1x + c1) +/- (a2x^2 + b2x + c2)
Add or subtract the coefficients of like terms: the x^2 coefficients, the x coefficients, and the constant terms are combined separately. The result is a new polynomial whose degree is at most the maximum degree of the inputs.
Worked Examples
Example 1: Adding Two Trinomials
Problem: Add (4x^2 - 3x + 7) and (2x^2 + 5x - 2).
Solution: Align like terms:\n x^2 terms: 4 + 2 = 6\n x terms: -3 + 5 = 2\n constants: 7 + (-2) = 5\nResult: 6x^2 + 2x + 5
Result: 6x^2 + 2x + 5 (degree 2, trinomial)
Example 2: Subtracting Polynomials with Cancellation
Problem: Subtract (3x^2 - x + 4) from (3x^2 + 2x - 1).
Solution: (3x^2 + 2x - 1) - (3x^2 - x + 4)\n= 3x^2 + 2x - 1 - 3x^2 + x - 4\n x^2 terms: 3 - 3 = 0 (cancel)\n x terms: 2 - (-1) = 3\n constants: -1 - 4 = -5\nResult: 3x - 5
Result: 3x - 5 (degree reduced from 2 to 1)
Frequently Asked Questions
What are the rules for adding polynomials together?
Adding polynomials follows the fundamental principle of combining like terms, which are terms with the same variable raised to the same power. To add two polynomials, you align terms by their degree and add their coefficients. For example, to add (3x^2 + 2x - 5) and (x^2 - 4x + 7), combine the x^2 terms (3 + 1 = 4), the x terms (2 + (-4) = -2), and the constants (-5 + 7 = 2) to get 4x^2 - 2x + 2. The degree of the result is always less than or equal to the maximum degree of the input polynomials. Addition is both commutative and associative.
Can adding or subtracting polynomials change the degree?
Yes, adding or subtracting polynomials can reduce the degree of the result when the leading terms cancel each other out. For example, adding (3x^2 + 2x + 1) and (-3x^2 + x - 4) gives 3x - 3, which has degree 1 instead of degree 2 because the x^2 terms sum to zero. Similarly, subtracting (x^2 + x) from (x^2 + 3) gives -x + 3, again reducing the degree. However, the degree can never increase beyond the maximum degree of the two input polynomials because no new higher-degree terms are created during addition or subtraction.
How do you organize polynomials before adding or subtracting them?
Before performing addition or subtraction, polynomials should be arranged in standard form, which means writing terms in descending order of their degree (highest power first). This organization makes it easy to visually align like terms for combining. Some textbooks use a vertical format where polynomials are stacked like column addition in arithmetic, with like terms vertically aligned. Others prefer horizontal format where parentheses clearly group each polynomial. If a polynomial is missing a term of a certain degree, you can insert a zero coefficient placeholder (like 0x) to maintain alignment during vertical addition.
What mistakes should you avoid when adding or subtracting polynomials?
The most frequent error is failing to distribute the negative sign to all terms when subtracting, leading to sign errors in the middle and constant terms. Another common mistake is combining unlike terms, such as adding x^2 and x coefficients together. Students also frequently drop terms entirely when rewriting expressions, especially constant terms or middle terms in longer polynomials. Arithmetic errors with negative numbers are another pitfall, particularly when a negative coefficient is being subtracted (resulting in a double negative that becomes positive). Always verify your result by substituting a test value like x = 1 into both the original expression and your answer.
What formula does Adding and Subtracting Polynomials Calculator use?
The formula used is described in the Formula section on this page. It is based on widely accepted standards in the relevant field. If you need a specific reference or citation, the References section provides links to authoritative sources.
Is my data stored or sent to a server?
No. All calculations run entirely in your browser using JavaScript. No data you enter is ever transmitted to any server or stored anywhere. Your inputs remain completely private.