To determine how many cubbies high the bookshelf will be, we need to create a rational expression using the height of the bookshelf and the height of each cubby hole.
Given:
- Height of the bookshelf: [tex]\( x^2 - 5x - 6 \)[/tex]
- Height of each cubby hole: [tex]\( x + 3 \)[/tex]
We need to form the rational expression:
[tex]\[ \frac{\text{Height of the bookshelf}}{\text{Height of each cubby hole}} \][/tex]
First, we need to simplify the numerator (the height of the bookshelf, [tex]\( x^2 - 5x - 6 \)[/tex]):
To do this, we factor the quadratic expression.
1. Factor the quadratic expression [tex]\( x^2 - 5x - 6 \)[/tex]:
To factor [tex]\( x^2 - 5x - 6 \)[/tex], we look for two numbers that multiply to [tex]\(-6\)[/tex] (the constant term) and add up to [tex]\(-5\)[/tex] (the coefficient of [tex]\(x\)[/tex]).
The numbers [tex]\(-6\)[/tex] and [tex]\(1\)[/tex] satisfy these conditions:
[tex]\[ x^2 - 5x - 6 = (x - 6)(x + 1) \][/tex]
Now, we have:
[tex]\[ x^2 - 5x - 6 = (x - 6)(x + 1) \][/tex]
2. Simplify the rational expression:
[tex]\[ \frac{x^2 - 5x - 6}{x + 3} \][/tex]
Substitute the factored form of the numerator:
[tex]\[ \frac{(x - 6)(x + 1)}{x + 3} \][/tex]
Now, observe whether the simplified form leads to cancellation:
In this case, no common factors between the numerator [tex]\((x - 6)(x + 1)\)[/tex] and the denominator [tex]\((x + 3)\)[/tex] are identified for cancellation.
Therefore, the rational expression representing how many cubbies high the bookshelf will be:
[tex]\[ \frac{(x - 6)(x + 1)}{x + 3} \][/tex]
Hence, this is our rational expression in its simplified form.
