Polynomial Arithmetic

Polynomial arithmetic refers to the operations performed on polynomials — mathematical expressions made up of variables raised to non-negative integer powers, combined with coefficients. Just as you can add and subtract ordinary numbers or fractions, you can add and subtract polynomials by combining the parts that look alike. This skill is foundational to nearly every branch of algebra, calculus, and applied mathematics. The key principle behind polynomial addition and subtraction is identifying and combining 'like terms' — terms that share the exact same variable(s) raised to the exact same power(s). For example, 3x² and −7x² are like terms because both involve x squared, so they can be combined into −4x². However, 3x² and 3x are not like terms because the exponents differ, so they cannot be merged into a single term. Mastering polynomial arithmetic builds the foundation for factoring, solving equations, working with rational expressions, and even calculus concepts like derivatives and integrals. The operations themselves are straightforward once you understand the structure of a polynomial and the rule of like terms.

Key takeaways

Identify Like Terms
Set Up the Operation (Horizontal or Vertical)
Distribute the Subtraction Sign (for Subtraction Only)
Combine Like Terms
Write the Result in Standard Form

Identify Like Terms

Like terms have identical variable parts, including the same variables and the same exponents. For instance, 5x³, −2x³, and x³ are all like terms. On the other hand, 4x²y and 4xy² are NOT like terms because the exponents on x and y differ. Scan both polynomials and mentally group or color-code terms that share the same variable part.

Set Up the Operation (Horizontal or Vertical)

You can arrange polynomial addition or subtraction either horizontally or in a vertical column format. The horizontal method writes both polynomials side by side: (3x² + 5x − 4) + (x² − 3x + 7). The vertical method stacks polynomials so like terms align in columns, similar to adding multi-digit numbers. Both methods yield the same result; choose whichever helps you stay organized.

Distribute the Subtraction Sign (for Subtraction Only)

When subtracting a polynomial, you must distribute the negative sign to every term in the polynomial being subtracted. This is the most common source of errors. For example: (5x² + 3x − 1) − (2x² − 4x + 6) becomes (5x² + 3x − 1) + (−2x² + 4x − 6) after distributing the minus sign. Every sign inside the second polynomial flips: positive terms become negative, and negative terms become positive.

Combine Like Terms

Add or subtract the coefficients of like terms, keeping the variable part unchanged. For example: 5x² + (−2x²) = 3x², then 3x + 4x = 7x, then −1 + (−6) = −7. Write the resulting terms together to form the answer polynomial. Conventionally, write the polynomial in standard form: descending order of degree (highest power first).

Write the Result in Standard Form

Standard form orders terms from the highest degree to the lowest. A polynomial like 4 + 3x − 2x³ + x² should be rewritten as −2x³ + x² + 3x + 4. The degree of the resulting polynomial is the highest exponent that appears. Double-check that no like terms remain uncombined, and verify all signs were handled correctly — especially after subtraction.

The Key Equation

(a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0) \pm (b_n x^n + b_{n-1}x^{n-1} + \cdots + b_1 x + b_0) = (a_n \pm b_n)x^n + (a_{n-1} \pm b_{n-1})x^{n-1} + \cdots + (a_1 \pm b_1)x + (a_0 \pm b_0)

a_k — Coefficient of the first polynomial (dimensionless): The numerical coefficient of the x^k term in the first polynomial P(x).
b_k — Coefficient of the second polynomial (dimensionless): The numerical coefficient of the x^k term in the second polynomial Q(x).
x^k — Monomial basis term (dimensionless): The variable x raised to the power k; terms with the same k are like terms and their coefficients combine directly.
n — Degree of the polynomial (dimensionless): The highest power of x with a nonzero coefficient; it determines the polynomial's degree and general shape.

Worked Example

Problem. Simplify: (4x³ − 2x² + 7x − 3) − (x³ + 5x² − 4x + 9)

3x^3 - 7x^2 + 11x - 12

Answer. 3x³ − 7x² + 11x − 12

Figure

Diagram

Origins · 1 minFrançois Viète and how this was discovered

François Viète

1540–1603 · French

François Viète was the first mathematician to use letters systematically to represent both known and unknown quantities in algebraic expressions, introducing what is now called symbolic algebra. His work 'In Artem Analyticem Isagoge' (1591) established the notation that made polynomial arithmetic with general coefficients possible, replacing the wordy rhetorical algebra of his predecessors. Viète was not a professional mathematician — he worked as a lawyer and royal counselor to King Henry III and Henry IV of France. He cracked a sophisticated Spanish cipher used during wartime, and the Spanish, convinced no human could break it, accused France of using sorcery.

Combining Like Terms — Polynomial addition and subtraction is built entirely on the ability to identify and combine like terms — terms with the same variable and exponent.
Definition and Degree of a Polynomial — Understanding what a polynomial is, how terms are structured, and what degree means is prerequisite knowledge for performing polynomial arithmetic.
Polynomial Multiplication — Once addition and subtraction are mastered, multiplication is the natural next operation; it uses the distributive property and then requires combining like terms.
The Distributive Property — Distributing a negative sign across a polynomial during subtraction is a direct application of the distributive property a(b + c) = ab + ac with a = −1.
Factoring Polynomials — Factoring is in many ways the reverse of polynomial arithmetic — recognizing how polynomials were added or expanded in order to break them back into simpler factors.