Question

box 1: dimensions: $x$ by $3x$ by $x^3$
area of base $= x(3x) = 3x^2$
box 2: dimensions: $x$ by $4x - 1$ by $x^3$
area of base $= x(4x - 1) = 4x^2 - x$

complete the statements about the number of terms in the polynomial representing the volume of each box.

• box 1s volume will be a $square$.
• box 2s volume will be a $square$.

explain your reasoning.
$square$

Explanation:

Step1: Calculate Box 1 volume

Volume = base area × height = $3x^2 \times x^3$
Simplify: $3x^{2+3}=3x^5$

Step2: Count Box1 volume terms

The polynomial $3x^5$ has 1 term.

Step3: Calculate Box 2 volume

Volume = base area × height = $(4x^2 - x) \times x^3$
Distribute: $4x^2 \times x^3 - x \times x^3 = 4x^{5} - x^{4}$

Step4: Count Box2 volume terms

The polynomial $4x^5 - x^4$ has 2 terms.

Answer:

Box 1's volume will be a monomial (1-term polynomial).
Box 2's volume will be a binomial (2-term polynomial).

Reasoning:

Box 1's volume simplifies to $3x^5$, which is a single term. Box 2's volume expands to $4x^5 - x^4$, which has two distinct, unlike terms that cannot be combined.