Multiplying Polynomials

Intuition and core definition

Multiplying polynomials applies the distributive property so every term of one factor multiplies every term of the other. Coefficients multiply and exponents on identical bases add. Like terms are combined only after all partial products are present.

Notation, language, and conditions

For $(\sum a_ix^i)(\sum b_jx^j)$, the coefficient of $x^k$ is the sum of products $a_ib_j$ with $i+j=k$. A grid, area model, or vertical layout organizes these pairings. The product of nonzero polynomials has degree equal to the sum of their degrees.

Why this idea matters

Polynomial multiplication distributes every term across every other term, and collecting like powers reveals the final coefficient structure.

A dependable method

1. Arrange each polynomial in descending powers and note missing degrees.
2. Multiply every term in the first polynomial by every term in the second.
3. Record signs, coefficient products, and exponent sums.
4. Group and combine like-degree partial products.
5. Check leading term, constant term, degree, and a numerical input.

Worked example

Common mistakes