Lesson: 12.2 Multiplication and Division with Rational Expressions

Erik is building a cubby bookshelf, which is divided into storage holes (cubbies) instead of shelves. He wants the height of the bookshelf to be [tex]x^2 - 5x - 6[/tex] and the width to be [tex]x^2 + 4x + 3[/tex]. Each cubby hole in the bookshelf will have a height of [tex]x + 3[/tex] and a width of [tex]x - 6[/tex].

Step 3 of 3: Multiply the rational expressions from Step 1 and Step 2 (and reduce to lowest terms) to obtain a rational expression that gives the total number of cubbies in the entire bookshelf.



Answer :

Let's find out the total number of cubbies Erik can make in the bookshelf. Given:

- The height of the bookshelf: [tex]\(x^2 - 5x - 6\)[/tex]
- The width of the bookshelf: [tex]\(x^2 + 4x + 3\)[/tex]
- The height of each cubby: [tex]\(x + 3\)[/tex]
- The width of each cubby: [tex]\(x - 6\)[/tex]

First, we need to determine the total area of the bookshelf. This is found by multiplying the height and width of the entire bookshelf:

[tex]\[ (x^2 - 5x - 6)(x^2 + 4x + 3) \][/tex]

Next, we determine the area of one cubby by multiplying its height and width:

[tex]\[ (x + 3)(x - 6) \][/tex]

To find the total number of cubbies, we must divide the total area of the bookshelf by the area of one cubby:

[tex]\[ \frac{(x^2 - 5x - 6)(x^2 + 4x + 3)}{(x + 3)(x - 6)} \][/tex]

By expanding and simplifying the expressions, we arrive at:

1. Expand [tex]\(x^2 - 5x - 6\)[/tex]:
[tex]\[ (x^2 - 5x - 6) = (x+1)(x-6) \][/tex]

2. Expand [tex]\(x^2 + 4x + 3\)[/tex]:
[tex]\[ (x^2 + 4x + 3) = (x+1)(x+3) \][/tex]

So,

[tex]\[ (x^2 - 5x - 6)(x^2 + 4x + 3) = (x+1)(x-6)(x+1)(x+3) \][/tex]

3. Expand [tex]\(x + 3\)[/tex] and [tex]\(x - 6\)[/tex]:
[tex]\[ (x + 3)(x - 6) \][/tex]

Thus,

[tex]\[ \frac{(x+1)(x-6)(x+1)(x+3)}{(x + 3)(x - 6)} \][/tex]

Upon canceling the common factors in the numerator and the denominator, we obtain:

[tex]\[ (x+1)(x+1) = (x+1)^2 \][/tex]

The simplified expression for the total number of cubbies in Erik's bookshelf is:

[tex]\[ x^2 + 2x + 1 \][/tex]