Question

select the word or number that correctly completes each sentence.

when simplified completely, the product of a monomial and a monomial is select a monomial.

when simplified completely, the product of a monomial and a binomial is select a trinomial.

when simplified completely, the product of a binomial and a binomial is select a monomial.

before being simplified completely, the product of a binomial and a trinomial has select terms.

Explanation:

Product of a monomial and a monomial

Using the Simplifying Monomials knowledge point

$$ (ax^m) \cdot (bx^n) = (ab)x^{m+n} $$

This result contains exactly one term, meaning it is always a monomial.

Product of a monomial and a binomial

Using the Distributive Property knowledge point

$$ a(b + c) = ab + ac $$

Multiplying a single term by two terms yields at most two terms. Therefore, it can never be a trinomial (which requires three terms).

Product of a binomial and a binomial

Using the Multiplying Binomials knowledge point

$$ (a + b)(c + d) = ac + ad + bc + bd $$

This product has at least two terms after simplification (e.g., \((x-1)(x+1) = x^2 - 1\)). Thus, it is never a monomial.

Product of a binomial and a trinomial before simplifying

Using the Distributive Property knowledge point

$$ (a + b)(c + d + e) = a(c + d + e) + b(c + d + e) = ac + ad + ae + bc + bd + be $$

This expansion results in exactly \(2 \times 3 = 6\) individual terms before any like terms are combined.

Answer:

Question 1

When simplified completely, the product of a monomial and a monomial is <blank>always</blank> a monomial.

Question 2

When simplified completely, the product of a monomial and a binomial is <blank>never</blank> a trinomial.

Question 3

When simplified completely, the product of a binomial and a binomial is <blank>never</blank> a monomial.

Question 4

Before being simplified completely, the product of a binomial and a trinomial has <blank>6</blank> terms.