Question

select the word or number that correctly completes each sentence.
when simplified completely, the product of a monomial and a monomial is a monomial.
when simplified completely, the product of a monomial and a binomial is a trinomial.
when simplified completely, the product of a binomial and a binomial is a monomial.
before being simplified completely, the product of a binomial and a trinomial has terms.

Explanation:

Step1: Define monomial/binomial/trinomial

A monomial is a single term: $ax^n$; binomial: $ax^n+by^m$; trinomial: $ax^n+by^m+cz^p$.

Step2: Monomial × Monomial

Multiply two single terms: $ax^n \times by^m = abx^ny^m$ (1 term, so always a monomial).

Step3: Monomial × Binomial

Distribute monomial: $c(ax+by)=acx+bcy$ (2 terms, so never a trinomial).

Step4: Binomial × Binomial

Multiply: $(a+b)(c+d)=ac+ad+bc+bd$ (4 terms before simplification; after simplification, it is never a monomial, unless terms cancel, but this is not guaranteed as a general rule).

Step5: Binomial × Trinomial (unsimplified)

Multiply each term: $(a+b)(c+d+e)=ac+ad+ae+bc+bd+be$ (6 distinct terms before simplification).

Answer:

1. When simplified completely, the product of a monomial and a monomial is always a monomial.
2. When simplified completely, the product of a monomial and a binomial is never a trinomial.
3. When simplified completely, the product of a binomial and a binomial is never a monomial.
4. Before being simplified completely, the product of a binomial and a trinomial has 6 terms.