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Adding and Subtracting Polynomials — Definition, Formula & Examples

Adding and subtracting polynomials means combining like terms — terms that share the same variable raised to the same power. You add or subtract the coefficients of like terms while keeping the variables and exponents unchanged.

Given two polynomials p(x)=∑aixip(x) = \sum a_i x^i and q(x)=∑bixiq(x) = \sum b_i x^i, their sum is p(x)+q(x)=∑(ai+bi)xip(x) + q(x) = \sum (a_i + b_i) x^i and their difference is p(x)−q(x)=∑(ai−bi)xip(x) - q(x) = \sum (a_i - b_i) x^i, where terms with matching degree are combined by adding or subtracting their coefficients.

How It Works

First, identify like terms across both polynomials — these are terms with the same variable and exponent, such as 3x23x^2 and −5x2-5x^2. For addition, combine their coefficients directly. For subtraction, distribute the negative sign to every term in the second polynomial, then combine like terms. Write the result in standard form, with terms ordered from highest degree to lowest.

Worked Example

Problem: Subtract: (4x3+2x2−7x+5)−(x3−3x2+2x−1)(4x^3 + 2x^2 - 7x + 5) - (x^3 - 3x^2 + 2x - 1)
Distribute the negative sign: Change the sign of every term in the second polynomial.
4x3+2x2−7x+5−x3+3x2−2x+14x^3 + 2x^2 - 7x + 5 - x^3 + 3x^2 - 2x + 1
Group like terms: Collect terms with matching powers of x.
(4x3−x3)+(2x2+3x2)+(−7x−2x)+(5+1)(4x^3 - x^3) + (2x^2 + 3x^2) + (-7x - 2x) + (5 + 1)
Combine coefficients: Add or subtract the coefficients for each group.
3x3+5x2−9x+63x^3 + 5x^2 - 9x + 6
Answer: 3x3+5x2−9x+63x^3 + 5x^2 - 9x + 6

Why It Matters

Adding and subtracting polynomials is foundational for factoring, solving polynomial equations, and simplifying rational expressions — skills you will use throughout Algebra 2, Precalculus, and Calculus. In physics and engineering, combining polynomial models often requires exactly this operation.

Common Mistakes

Mistake: Forgetting to distribute the negative sign to every term when subtracting
Correction: When you subtract a polynomial, the minus sign applies to all its terms, not just the first one. Write it out: −(x3−3x2)-(x^3 - 3x^2) becomes −x3+3x2-x^3 + 3x^2, not −x3−3x2-x^3 - 3x^2.