Polynomial Operations

Take the number 374 and break it into pieces: 3 \times 100 + 7 \times 10 + 4 \times 1. That is 3 \times 10^2 + 7 \times 10^1 + 4 \times 10^0. Now replace 10 with a letter — call it x — and you have 3x^2 + 7x + 4. A polynomial.

When you add two three-digit numbers, you line up the hundreds, tens, and units and combine matching columns. Polynomial addition does the same thing — you line up terms with matching powers of x and combine them. When you multiply two numbers using the long-multiplication grid, every digit of the first number multiplies every digit of the second. Polynomial multiplication works identically — every term of the first polynomial multiplies every term of the second.

The operations are not metaphorically similar. They are structurally the same. This article builds each one from that parallel, so the rules feel familiar rather than new.

Addition and subtraction

Adding two polynomials is a matter of collecting like terms — terms that have the same power of x. Take p(x) = 3x^3 + 5x^2 - 2x + 1 and q(x) = x^3 - 4x^2 + 7. To compute p(x) + q(x), align the terms by power:

Each power of x is an independent "column" — the x^3 terms combine with x^3 terms, the x^2 with x^2, and so on. If one polynomial is missing a power (here, q has no x term), you treat its coefficient as 0.

Subtraction works the same way, but you flip the signs of the second polynomial first. To compute p(x) - q(x):

The sign flip is the only difference. Once you have flipped, you are adding again.

A key observation: the degree of the sum is at most the highest degree among the inputs. It could be less — if the leading terms cancel (like adding 2x^3 and -2x^3), the degree drops. But it can never be higher. Addition and subtraction never create new powers of x.

Multiplication

Multiplication does create new powers. When you multiply a term ax^m by a term bx^n, the result is abx^{m+n}. The coefficients multiply; the exponents add. This is just the exponent law x^m \cdot x^n = x^{m+n} applied to each pair of terms.

To multiply two polynomials, distribute every term of the first across every term of the second. Take p(x) = 2x + 3 and q(x) = x^2 - x + 4.

Expand each part:

Now collect like terms:

The degree of the product is 1 + 2 = 3 — the sum of the degrees of the two factors. This always holds: if p has degree m and q has degree n, the product p \cdot q has degree m + n, because the highest-power terms a_m x^m and b_n x^n multiply to give a_m b_n x^{m+n}, and nothing else in the product reaches that power.

Division: the long-division algorithm

Dividing a polynomial p(x) by a polynomial d(x) is like long division with numbers. You ask: "How many times does d(x) go into p(x)?" — and you get a quotient q(x) and a remainder r(x) such that

with the degree of r(x) strictly less than the degree of d(x). If the remainder is zero, d(x) divides p(x) exactly.

The algorithm works by repeatedly removing the leading term. Here is the recipe:

1. Look at the leading term of p(x) and the leading term of d(x). Divide them to get the first term of q(x).
2. Multiply d(x) by that term and subtract the result from p(x).
3. What remains is a new polynomial of lower degree. Repeat from step 1 with this new polynomial, until the remainder has degree less than d(x).

Take p(x) = 2x^3 + 3x^2 - 5x + 6 divided by d(x) = x - 2.

Round 1. Leading terms: 2x^3 \div x = 2x^2. Multiply: (x - 2)(2x^2) = 2x^3 - 4x^2. Subtract: (2x^3 + 3x^2 - 5x + 6) - (2x^3 - 4x^2) = 7x^2 - 5x + 6.

Round 2. Leading terms: 7x^2 \div x = 7x. Multiply: (x - 2)(7x) = 7x^2 - 14x. Subtract: (7x^2 - 5x + 6) - (7x^2 - 14x) = 9x + 6.

Round 3. Leading terms: 9x \div x = 9. Multiply: (x - 2)(9) = 9x - 18. Subtract: (9x + 6) - (9x - 18) = 24.

The remainder is 24 (degree 0, less than the degree of d(x) = x - 2). The quotient is 2x^2 + 7x + 9.

So: 2x^3 + 3x^2 - 5x + 6 = (x - 2)(2x^2 + 7x + 9) + 24.

You can verify by expanding the right side: (x - 2)(2x^2 + 7x + 9) = 2x^3 + 7x^2 + 9x - 4x^2 - 14x - 18 = 2x^3 + 3x^2 - 5x - 18, and -18 + 24 = 6. The original polynomial is recovered.

The Remainder Theorem

Look at the division you just did. The remainder was 24. Now evaluate the dividend at x = 2 (the root of the divisor x - 2):

The remainder and the evaluation give the same number. This is not a coincidence.

The proof is one line. By the division identity, p(x) = (x - a) \cdot q(x) + r, where r is a constant (because the remainder's degree must be less than 1). Substitute x = a: p(a) = (a - a) \cdot q(a) + r = 0 + r = r. The remainder equals p(a).

This is a powerful shortcut. If someone asks "what is the remainder when x^{100} - 2x + 5 is divided by x - 1?", you do not need long division on a degree-100 polynomial. Just evaluate at x = 1: the remainder is 1 - 2 + 5 = 4.

The Factor Theorem

The Factor Theorem is the special case of the Remainder Theorem where the remainder is zero.

The "if" direction: if p(a) = 0, then the Remainder Theorem says the remainder when dividing by (x - a) is 0, so (x - a) divides p(x) exactly.

The "only if" direction: if (x - a) divides p(x), then p(x) = (x - a) \cdot q(x), and substituting x = a gives p(a) = 0.

So testing whether (x - a) is a factor of p(x) reduces to a single evaluation. Take p(x) = x^3 - 6x^2 + 11x - 6. Is (x - 1) a factor? Check: p(1) = 1 - 6 + 11 - 6 = 0. Yes. Is (x - 4) a factor? Check: p(4) = 64 - 96 + 44 - 6 = 6 \neq 0. No.

The Factor Theorem is the bridge between polynomial operations and Polynomial Factorization. Once you know that p(a) = 0, you know (x - a) is a factor, and you can use long division to find the remaining quotient. Then repeat: test the quotient for more roots, and keep peeling off factors until the polynomial is fully broken down.

Two worked examples

Common confusions

• "You can add 3x^2 and 5x because they both have x." No — like terms must have the same power of x. The terms 3x^2 and 5x are different species. 3x^2 + 5x cannot be simplified further, just as you cannot add metres and square metres.

• "The degree of (x^2 + 1)(x^3 + x) is 5... or is it 3?" It is 5. The degree of a product is the sum of the degrees: 2 + 3 = 5. The leading term of the product is x^2 \cdot x^3 = x^5.

• "When subtracting polynomials, only the first term of the second polynomial gets its sign flipped." Every term of the second polynomial gets its sign flipped — the subtraction distributes across the entire polynomial, not just the leading term. Forgetting to flip the sign of the constant term is one of the most common errors.

• "If p(a) = 0, then (x - a) is a factor of p(x) — but only for integer a." The Factor Theorem works for any value of a, integer or not. If p(\tfrac{1}{2}) = 0, then (x - \tfrac{1}{2}) is a factor. You can also write this as (2x - 1) being a factor, after clearing the fraction.

• "Division of polynomials always has remainder zero." Not at all — most divisions leave a remainder, just like 17 \div 3 leaves remainder 2 in integer arithmetic. The remainder is zero only when the divisor is an exact factor.

• "The Remainder Theorem works only when dividing by a linear polynomial." In its standard form, yes — the theorem is about dividing by (x - a). For higher-degree divisors, you still get a remainder, but it is a polynomial of degree less than the divisor, not a single number.

Going deeper

Polynomials form a ring

The set of polynomials with real coefficients, written \mathbb{R}[x], obeys the same structural rules as the integers. Addition is commutative and associative; multiplication is commutative, associative, and distributes over addition. There is an additive identity (the zero polynomial) and a multiplicative identity (the constant 1). But most polynomials have no multiplicative inverse that is itself a polynomial — the inverse of x would have to satisfy x \cdot (?) = 1, and no polynomial in x does that. So \mathbb{R}[x] is a ring, not a field — the same distinction that separates \mathbb{Z} from \mathbb{Q}.

The division algorithm you learned above is the polynomial analogue of the integer division algorithm (Euclid's algorithm). It produces a unique quotient and remainder, and that uniqueness is the foundation for everything from the Factor Theorem to the construction of polynomial GCDs.

Synthetic division — a shortcut for linear divisors

When the divisor is linear — (x - a) — the long-division algorithm simplifies dramatically. Instead of writing out the full subtraction at each step, you can track only the coefficients and use a compact table. This shortcut is called synthetic division. For the division of 2x^3 + 3x^2 - 5x + 6 by (x - 2) from earlier, the synthetic layout uses only the coefficients [2, 3, -5, 6] and the value a = 2:

Start with 2. Multiply by a = 2 to get 4; add to 3 to get 7. Multiply 7 by 2 to get 14; add to -5 to get 9. Multiply 9 by 2 to get 18; add to 6 to get 24. The bottom row reads [2, 7, 9, 24] — the first three entries are the quotient coefficients (2x^2 + 7x + 9) and the last is the remainder (24). The same answer, with much less writing.

The degree of the remainder bounds the information

When you divide by a degree-d polynomial, the remainder has degree at most d - 1. This is not an accident — it is the stopping condition of the algorithm, and it is what makes the quotient and remainder unique. If you allowed the remainder to have the same degree as the divisor, you could subtract one more copy and reduce further. The algorithm stops precisely when it can go no further.

This bound also explains why the Remainder Theorem, as stated, only gives a single number for a linear divisor. The remainder when dividing by a degree-1 polynomial has degree at most 0 — it is a constant. For a degree-2 divisor, the remainder would be a linear polynomial ax + b, determined by two values rather than one.

Where this leads next

The four operations you learned here are the grammar of every polynomial chapter that follows.

• Polynomials — Introduction — the definitions, terminology, and examples that set up everything in this article.
• Polynomial Factorization — using the Factor Theorem and other techniques to break polynomials into irreducible pieces.
• Algebraic Identities — the standard identities (a + b)^2, (a - b)^2, a^2 - b^2 and their cubic extensions, used everywhere in multiplication and factorisation.
• Quadratic Equations — Introduction — where the Factor Theorem meets the quadratic formula, and polynomial roots become the center of the story.
• Algebraic Expressions — the broader family of expressions that includes polynomials as a special case.