Question

polynomials and factoring — review assignment

1. given the rectangle below with the area as shown, find:

(a) the width of the rectangle.
(b) the perimeter of the rectangle.

$3x^2 - 14x - 5$
$x - 5$
width

Explanation:

Part (a)

Step1: Recall area of rectangle

The area \( A \) of a rectangle is given by \( A = \text{length} \times \text{width} \). Here, the area is \( 3x^2 - 14x - 5 \) and one side (length) is \( x - 5 \). So, to find the width, we divide the area by the length: \( \text{width} = \frac{3x^2 - 14x - 5}{x - 5} \).

Step2: Factor the quadratic

Factor \( 3x^2 - 14x - 5 \). We need two numbers that multiply to \( 3 \times (-5) = -15 \) and add to \( -14 \). Those numbers are \( -15 \) and \( 1 \). So,

$$
LATEXBLOCK0
$$

Step3: Divide by \( x - 5 \)

Now, divide \( (3x + 1)(x - 5) \) by \( x - 5 \) (assuming \( x
eq 5 \)). Canceling out \( x - 5 \), we get the width as \( 3x + 1 \).

Step1: Recall perimeter formula

The perimeter \( P \) of a rectangle is \( P = 2(\text{length} + \text{width}) \). We know the length is \( x - 5 \) and the width is \( 3x + 1 \) (from part (a)).

Step2: Find length + width

First, calculate \( (x - 5) + (3x + 1) \):

$$ (x - 5) + (3x + 1) = x - 5 + 3x + 1 = 4x - 4 $$

Step3: Calculate perimeter

Multiply by 2: \( P = 2(4x - 4) = 8x - 8 \).

Answer:

(a):
The width of the rectangle is \( 3x + 1 \).

Part (b)