Question

question 8
if the area of a rectangle is represented by $x^{2}+8x + 15$ and its length is represented by $x + 5$, which expression represents the width of the rectangle?
$x + 3$
$x - 3$
$x^{2}+6x + 5$
$x^{2}+7x + 10$

Explanation:

Step1: Recall the formula for the area of a rectangle

The area of a rectangle \(A = l\times w\), where \(A\) is the area, \(l\) is the length, and \(w\) is the width. So, \(w=\frac{A}{l}\). Given \(A = x^{2}+8x + 15\) and \(l=x + 5\), we need to perform polynomial division or factorization.

Step2: Factor the quadratic expression

Factor \(x^{2}+8x + 15\). We look for two numbers that multiply to \(15\) and add up to \(8\). The numbers are \(3\) and \(5\). So, \(x^{2}+8x + 15=(x + 3)(x + 5)\) using the formula \(x^{2}+(a + b)x+ab=(x + a)(x + b)\) where \(a = 3\) and \(b = 5\).

Step3: Calculate the width

Since \(w=\frac{A}{l}\) and \(A=(x + 3)(x + 5)\), \(l=x + 5\), then \(w=\frac{(x + 3)(x + 5)}{x + 5}\). Cancel out the common factor \((x + 5)\) (assuming \(x
eq - 5\)), and we get \(w=x + 3\).

Answer:

\(x + 3\)