45.3 Polynomial-by-Polynomial Partial Products

Polynomial-by-Polynomial Partial Products is the process of multiplying two multi-term polynomials together by selecting each term of the first polynomial one at a time and distributing it fully across every term of the second polynomial, generating a complete collection of partial products before any combining takes place.

First Polynomial Term Selection

Procedure

A single term is selected from the first polynomial factor to be distributed across the entire second polynomial.

Example

For $(2x+3)(x-4)$, the term $2x$ is selected first from the first polynomial factor.

Selected Term Distribution across the Second Polynomial

Procedure

The selected term is treated as a monomial being multiplied across every term of the second polynomial, using the same distribution process established for monomial-by-polynomial multiplication.

Example

The selected term $2x$ is distributed across both $x$ and $-4$ from the second polynomial.

First Partial Polynomial Product

Procedure

The result of distributing the first selected term across the entire second polynomial is recorded as the first partial product.

Example

Distributing $2x$ across $x-4$ produces the first partial product:

Next Polynomial Term Selection

Procedure

The next term from the first polynomial factor is selected, continuing the process from where the previous selection left off.

Example

The term $3$ is selected next from the first polynomial factor.

Additional Partial Polynomial Product

Procedure

The newly selected term is distributed across the entire second polynomial, producing another partial product, exactly as with the first selected term.

Example

Distributing $3$ across $x-4$ produces the second partial product:

Complete Factor-Term Pair Coverage

Requirement

This selection-and-distribution cycle is repeated for every term in the first polynomial factor, without exception, ensuring every possible pairing between a term of the first factor and a term of the second factor has been generated.

Confirming Completeness

For a first factor with two terms and a second factor with two terms, exactly four total term pairings should exist across all partial products, confirming complete coverage.

Partial Product Sign Preservation

Requirement

Each partial product retains the sign determined by combining the sign of its selected term with the sign of each term in the second polynomial it was multiplied against.

Example

The second partial product, $3x-12$, preserves a negative sign on its second term, since the positive $3$ was multiplied against the negative $-4$.

Combined Partial Product Expression

Procedure

All of the individual partial products generated from every term selection are written together as a single combined expression, ready for the subsequent step of combining like terms.

Example

Combining both partial products from the running example produces:

representing the complete, unsimplified collection of partial products before any like terms are combined.