Question

1. which polynomial represents the area of a rectangle with sides $3x$ and $x^2 + 2x + 1$?

\bigcirc a. $3x^3 + 6x^2 + 3x$
\bigcirc b. $3x^2 + 2x + 1$
\bigcirc c. $3x^2 + 5x + 3$
\bigcirc d. $3x^3 + 6x^2 - 3x$

Explanation:

Step1: Recall area of rectangle formula

The area \( A \) of a rectangle is given by the product of its length and width, i.e., \( A = \text{length} \times \text{width} \). Here, the sides are \( 3x \) and \( x^2 + 2x + 1 \), so we need to compute \( 3x \times (x^2 + 2x + 1) \).

Step2: Distribute \( 3x \) over the polynomial

Using the distributive property (also known as the distributive law of multiplication over addition), we multiply \( 3x \) with each term inside the parentheses:
\[

$$\begin{align*} 3x \times (x^2 + 2x + 1) &= 3x \times x^2 + 3x \times 2x + 3x \times 1 \\ &= 3x^{1 + 2} + 6x^{1 + 1} + 3x \\ &= 3x^3 + 6x^2 + 3x \end{align*}$$

\]

Answer:

a. \( 3x^3 + 6x^2 + 3x \)