Question

multiplying polynomials: mastery test
image of a grid with blue rectangles labeled with terms like x and 1
options:
a. (x + 1)(x + 3)
b. x(x + 3)
c. 3(x + 1)
d. (3x + 1)(x + 1)

Explanation:

Step1: Analyze the area model

The rectangle is divided into parts. Let's find the length and width of the overall rectangle. Looking at the columns: one column has terms involving \(x\) and constants, and the other? Wait, no, let's count the number of \(x\) terms and constant terms. Wait, actually, let's look at the rows and columns. The vertical side (height) has terms: let's see the left part (width) and bottom part (height). Wait, the horizontal (length) of the big rectangle: let's see the number of \(x\) terms and the constant. Wait, the left column (width) has \(x + x + x+?\) Wait, no, looking at the model, the horizontal length (top - bottom) has \(x\) terms: three \(x\)s? Wait, no, the vertical columns: one column is \(x\) (wait, no, the first column (left) has a big square? Wait, no, maybe I misread. Wait, the model: let's see the horizontal (length) and vertical (width) of the rectangle. Let's count the number of \(x\) terms and constant terms. The horizontal side (length) can be broken into \((3x + 1)\) and vertical side (width) into \((x + 1)\)? Wait, no, let's check the options. Option D is \((3x + 1)(x + 1)\). Let's verify:

Multiply \((3x + 1)(x + 1)\) using distributive property (FOIL or area model). The area model would have four parts? Wait, no, the given model has 6 parts? Wait, maybe the model is a rectangle with length \((3x + 1)\) and width \((x + 1)\). Let's expand \((3x + 1)(x + 1)\):

\(3x \times x + 3x \times 1 + 1 \times x + 1 \times 1 = 3x^2 + 3x + x + 1 = 3x^2 + 4x + 1\). Wait, but the model: let's see the parts. The left column (width) has a big square? Wait, maybe the model is structured as: the vertical (height) is \((x + 1)\) and horizontal (length) is \((3x + 1)\). Let's check the options. Option D is \((3x + 1)(x + 1)\). Let's check other options:

Option A: \((x + 1)(x + 3)\) expands to \(x^2 + 4x + 3\).

Option B: \(x(x + 3)=x^2 + 3x\).

Option C: \(3(x + 1)=3x + 3\).

Now, looking at the area model, there are terms with \(x^2\), \(x\), and constants. Wait, maybe the big square is \(x^2\)? No, wait, the first row (top) has a big rectangle (maybe \(x \times x\)? No, maybe the left part (width) has \(3x\) (three \(x\) terms) and 1, and the bottom (height) has \(x\) and 1. So the length is \(3x + 1\) and width is \(x + 1\), so the product is \((3x + 1)(x + 1)\), which is option D.

Answer:

D. \((3x + 1)(x + 1)\)