45.2 Monomial-by-Polynomial Multiplication

Monomial-by-Polynomial Multiplication is the process of multiplying a single monomial by a polynomial containing multiple terms, by distributing the monomial across every term of the polynomial individually, combining coefficients and adding exponents of matching bases at each step.

Exterior Monomial Factor Identification

Procedure

The single monomial factor standing outside the polynomial is identified separately, noting both its numerical coefficient and its variable factors with their exponents.

Example

In $3x^2(4x-5)$, the exterior monomial factor is $3x^2$.

Polynomial Term Inventory

Procedure

Every individual term within the polynomial factor, along with its own sign, is listed separately, forming a complete inventory to be multiplied by the exterior monomial.

Example

For the same expression, the polynomial term inventory consists of $4x$ and $-5$.

Monomial Distribution to Every Term

Procedure

The exterior monomial is multiplied against each term in the inventory individually, one at a time, ensuring no term is skipped.

Polynomial-Factor Coefficient Multiplication

Procedure

For each term pairing, the numerical coefficient of the exterior monomial is multiplied by the numerical coefficient of the polynomial term.

Example

For the pairing of $3$ and $4$, the coefficients multiply to $12$.

Common-Base Exponent Addition

Procedure

For each term pairing, the exponents of any matching variable bases are added together, following the product rule for exponents.

Example

Multiplying $x^2$ by $x$, understood as $x^1$, produces $x^3$, since $2+1=3$.

Distributed Product Sign Determination

Procedure

The sign of each resulting term is determined by combining the sign of the exterior monomial with the sign of the specific polynomial term being multiplied against it.

Example

Multiplying the positive monomial $3x^2$ by the negative term $-5$ produces a negative result.

Distributed Term Collection

Procedure

Every individual product resulting from the distribution is collected together, forming the complete unsimplified result of the multiplication.

Example

Distributing $3x^2$ across $4x-5$ produces:

Monomial-Polynomial Product Standardization

Procedure

The collected products are checked for any like terms that can be combined, and the final result is arranged in standard form, ordered by descending degree.

Example

Since $12x^3$ and $-15x^2$ are already unlike terms with no further combination possible, and already ordered by descending degree, the final standardized product is: