Add and Subtract Polynomials Calculator

Understanding Polynomial Addition and Subtraction

Polynomials are algebraic expressions composed of variables, coefficients, and exponents combined using addition, subtraction, and multiplication. Adding and subtracting polynomials is one of the most fundamental operations in algebra. The core principle behind both operations is combining like terms, which means adding or subtracting the numerical coefficients of terms that have the exact same variable raised to the exact same power.

Whether working with linear polynomials, quadratics, cubics, or higher-order expressions, this online calculator helps you quickly compute the sum or difference, automatically arrange terms in standard descending form, and evaluate the resulting polynomial at any specified value. For expanding products of algebraic expressions, explore our Expand Polynomials Calculator and Quadratic Calculator.

The Standard Form of a Polynomial

A polynomial in a single variable $x$ is expressed in standard form when its terms are arranged in descending order of degree:

$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$

Here, $n$ represents the non-negative integer degree of the polynomial, $a_n \neq 0$ is the leading coefficient, and $a_0$ is the constant term. When combining two polynomials $P(x)$ and $Q(x)$, terms corresponding to degree $k$ are paired together:

$$(P \pm Q)(x) = \sum_{k=0}^{\max(n, m)} (a_k \pm b_k) x^k$$

Step-by-Step Method: Adding Polynomials

To add two polynomials $P(x)$ and $Q(x)$:

1. Enclose both polynomials: Write the addition expression $(P(x)) + (Q(x))$.
2. Remove parentheses: Since addition preserves all signs, rewrite the terms directly without parentheses.
3. Group like terms: Place terms with identical powers adjacent to each other.
4. Combine coefficients: Add the numerical coefficients for each shared degree.
5. Order terms: Write the simplified expression in descending order of exponents.

Step-by-Step Method: Subtracting Polynomials

Subtraction requires one extra critical step: distributing the negative sign to every term of the subtrahend polynomial:

1. Enclose both polynomials: Write $(P(x)) - (Q(x))$.
2. Distribute the minus sign: Multiply each term inside $Q(x)$ by $-1$, flipping positive terms to negative and negative terms to positive.
3. Group and combine: Group like powers together and add the resulting coefficients.
4. Final verification: Check that all like terms have been combined and no redundant powers remain.

Worked Example

Let $P(x) = 4x^3 - 2x^2 + 5x - 7$ and $Q(x) = 2x^3 + 6x^2 - 8x + 3$.

1. Addition $P(x) + Q(x)$:

$$(4x^3 - 2x^2 + 5x - 7) + (2x^3 + 6x^2 - 8x + 3)$$ $$= (4+2)x^3 + (-2+6)x^2 + (5-8)x + (-7+3)$$ $$= 6x^3 + 4x^2 - 3x - 4$$

2. Subtraction $P(x) - Q(x)$:

$$(4x^3 - 2x^2 + 5x - 7) - (2x^3 + 6x^2 - 8x + 3)$$ $$= 4x^3 - 2x^2 + 5x - 7 - 2x^3 - 6x^2 + 8x - 3$$ $$= (4-2)x^3 + (-2-6)x^2 + (5+8)x + (-7-3)$$ $$= 2x^3 - 8x^2 + 13x - 10$$