Question

the area of a rectangular rug is given by the trinomial ( r^{2}-3r - 10 ). what are the possible dimensions of the rug? use factoring. the length and the width of the rectangle are (use a comma to separate answers as needed.)

Explanation:

Step1: Factor the trinomial

We have the trinomial \(r^{2}-3r - 10\).
We need to find two numbers \(a\) and \(b\) such that \(a\times b=-10\) and \(a + b=-3\).
The numbers are \(a = 2\) and \(b=-5\) since \(2\times(-5)=-10\) and \(2+(-5)=-3\).
Using the formula \(x^{2}+(a + b)x+ab=(x + a)(x + b)\), for \(x = r\), \(a = 2\), \(b=-5\), we get \(r^{2}-3r-10=(r + 2)(r-5)\)

Step2: Relate to rectangle dimensions

The area of a rectangle is \(A=\text{length}\times\text{width}\). Given \(A=r^{2}-3r - 10=(r + 2)(r - 5)\)

Answer:

\(r + 2,r-5\)