Numbers have addition, subtraction, multiplication, and division — and so do polynomials. The same four operations that govern ordinary arithmetic extend naturally to polynomial expressions, each with its own set of rules governing how terms interact, how degrees change, and what form the result takes.
Some of these operations are simple: adding polynomials amounts to collecting like terms. Others demand more care — polynomial long division mirrors its numerical counterpart but requires tracking multiple terms across multiple steps. Together, these operations provide the machinery behind factoring, solving polynomial equations, and simplifying complex algebraic expressions.
Adding Polynomials
Adding two polynomials reduces to a single principle: combine like terms. Two terms are like terms when they share the same variable raised to the same exponent — 3x2 and 5x2 are like terms, while 3x2 and 5x3 are not.
Given (2x2+3x−1)+(x2−5x+4), group the terms by degree: the x2 terms are 2x2 and x2, the x terms are 3x and −5x, and the constants are −1 and 4. Adding each group gives 3x2−2x+3.
The process works identically for any number of terms and any degree. Align terms by matching exponents, add their coefficients, and write the result. Terms with no counterpart in the other polynomial pass through unchanged — adding x3+2x and 3x2−x+5 produces x3+3x2+x+5, where x3 and 3x2 each carried over directly.
The sum of two polynomials is always a polynomial. Its degree is at most the larger of the two input degrees — and in rare cases it can be less, if the leading terms cancel. Adding 2x3+x−1 and −2x3+4x+7 eliminates the cubic terms entirely, leaving 5x+6, a polynomial of degree 1.
Subtracting Polynomials
Subtraction follows the same logic as addition, with one extra step: distribute the negative sign across every term of the polynomial being subtracted before combining like terms.
For (2x2+3x−1)−(x2−5x+4), first negate the second polynomial: −x2+5x−4. Then add: (2x2+3x−1)+(−x2+5x−4)=x2+8x−5. Every term inside the parentheses changes sign — including terms that were already negative. The −5x becomes +5x and the +4 becomes −4.
This is where most errors occur. Negating only the first term of the subtracted polynomial while leaving the rest untouched is the single most common mistake in polynomial subtraction. The expression (5x2+2x−3)−(x2−4x+1) is sometimes incorrectly computed as 4x2−2x−2 when only x2 gets negated. The correct result is 4x2+6x−4.
Like addition, subtraction always produces a polynomial. The degree of the result is at most the larger of the two input degrees, with the same possibility of cancellation reducing it further.
Multiplying Polynomials
Multiplication requires every term of one polynomial to be multiplied by every term of the other. For each pair, multiply the coefficients and add the exponents on matching variables, then combine like terms in the resulting expression.
For (x+2)(x2−3x+1), distribute each term of the binomial across the trinomial. The term x produces x⋅x2=x3, x⋅(−3x)=−3x2, and x⋅1=x. The term 2 produces 2⋅x2=2x2, 2⋅(−3x)=−6x, and 2⋅1=2. Collecting everything: x3−3x2+x+2x2−6x+2. Combining like terms gives x3−x2−5x+2.
The degree of a product equals the sum of the degrees of the factors — always, without exception. A degree-2 polynomial times a degree-3 polynomial produces a degree-5 polynomial. This follows from the fact that the highest-degree terms in each factor multiply to produce the highest-degree term in the product, and no other pair of terms can reach that degree.
The number of terms before combining can grow quickly. Multiplying an m-term polynomial by an n-term polynomial generates m×n individual products. A binomial times a trinomial produces 6 terms before simplification; two four-term polynomials produce 16. Careful bookkeeping — and a systematic approach to distributing — prevents missed terms.
Special Products
Certain multiplication patterns appear so frequently that recognizing them on sight saves significant work. Each is a direct consequence of distribution, but memorizing the results avoids repeating the same steps every time.
Squaring a binomial produces a trinomial. The sum (a+b)2 expands to a2+2ab+b2, and the difference (a−b)2 expands to a2−2ab+b2. The middle term is always twice the product of the two original terms — a detail that connects directly to identifying perfect square trinomials when factoring.
Multiplying conjugate binomials — a sum and a difference of the same two terms — eliminates the middle term entirely. The product (a+b)(a−b) equals a2−b2, a clean difference of squares. No linear term survives because the outer and inner products cancel.
The cubic identities extend the same logic one degree further:
(a+b)(a2−ab+b2)=a3+b3
(a−b)(a2+ab+b2)=a3−b3
Each pairs a linear factor with a quadratic factor to produce a two-term cubic. These patterns are the multiplication side of the sum and difference of cubes formulas used in factoring — the same identities read in the opposite direction.
FOIL Method
FOIL is a mnemonic for multiplying two binomials: First, Outer, Inner, Last. It names the four products that arise when each term of one binomial is distributed across the other.
For (2x+3)(x−4), the four steps produce: First 2xcdotx=2x2, Outer 2xcdot(−4)=−8x, Inner 3cdotx=3x, Last 3cdot(−4)=−12. Combining the middle terms gives 2x2−5x−12.
FOIL is nothing more than a structured way to ensure all four products are accounted for — it is a special case of the general distribution used in multiplying any two polynomials. Its value lies in speed and reliability for the specific case of two binomials, where exactly four products always arise.
The method does not extend beyond binomials. Attempting to apply FOIL to a binomial times a trinomial, or to two trinomials, misses terms. A binomial times a trinomial requires six products, not four. For anything beyond two binomials, the full term-by-term distribution described in the previous section is the correct approach.
Polynomial Long Division
Dividing one polynomial by another follows a process that mirrors numerical long division. The idea is to find a quotient and a remainder that satisfy:
dividend=(divisor×quotient)+remainder
To divide x3−2x2+5x−3 by x−1, begin by dividing the leading term of the dividend by the leading term of the divisor: x3÷x=x2. Multiply x2 by the entire divisor: x2(x−1)=x3−x2. Subtract this from the dividend to get −x2+5x−3. Repeat: −x2÷x=−x, multiply −x(x−1)=−x2+x, subtract to get 4x−3. One more pass: 4x÷x=4, multiply 4(x−1)=4x−4, subtract to get 1. The quotient is x2−x+4 with remainder 1.
The process terminates when the degree of what remains is less than the degree of the divisor. If the remainder is zero, the divisor divides the dividend evenly — meaning the divisor is a factor of the dividend.
Missing powers in the dividend require placeholder terms. Dividing x3+4 by x+1 requires writing the dividend as x3+0x2+0x+4 to keep terms aligned through each subtraction step.
Synthetic Division
When the divisor is a linear binomial of the form (x−c), synthetic division offers a streamlined alternative to long division. It uses only the coefficients of the dividend and the value c, reducing the process to a compact sequence of multiplications and additions.
To divide x3−2x2+5x−3 by (x−1), write the coefficients 1,−2,5,−3 in a row and place c=1 to the left. Bring down the first coefficient: 1. Multiply by c: 1cdot1=1. Add to the next coefficient: −2+1=−1. Multiply by c: −1cdot1=−1. Add: 5+(−1)=4. Multiply: 4cdot1=4. Add: −3+4=1. The bottom row reads 1,−1,4,1 — giving a quotient of x2−x+4 with remainder 1, matching the long division result exactly.
The connection to polynomial evaluation is immediate. The remainder produced by synthetic division equals P(c), the value of the dividend evaluated at x=c. This is the Remainder Theorem in action, and it means synthetic division doubles as a fast way to compute P(c) without substituting into every term. If P(c)=0, then (x−c) divides the polynomial evenly and c is a root.
The limitation is strict: synthetic division works only for divisors of the form (x−c). Dividing by x2+1, by 2x−3, or by any polynomial of degree two or higher requires the full long division procedure.
Degree and Operations
Each arithmetic operation affects the degree of the result in a specific, predictable way. These rules serve as both a guide during computation and a check after it.
For addition and subtraction, the degree of the result is at most the maximum of the two input degrees. If P(x) has degree 4 and Q(x) has degree 3, then P(x)+Q(x) has degree at most 4. The "at most" matters — when the leading terms cancel, the degree drops. Adding 2x3+x and −2x3+5x2 produces 5x2+x, a degree-2 result from two degree-3 inputs.
For multiplication, the degree of the product equals the sum of the degrees of the factors, with no exceptions. A degree-2 polynomial times a degree-3 polynomial always produces a degree-5 polynomial. Leading coefficients multiply but never cancel, since neither is zero.
For division, the degree of the quotient equals the degree of the dividend minus the degree of the divisor. Dividing a degree-5 polynomial by a degree-2 polynomial produces a degree-3 quotient. The remainder, if nonzero, has degree strictly less than the divisor.
These relationships provide a quick sanity check. If multiplying two quadratics yields a degree-3 result, something went wrong. If dividing a cubic by a linear polynomial produces a cubic quotient, a step was missed.