Question

The length of a rectangle is 20 units more than its width. The area of the rectangle is x4−100.

Which expression represents the width of the rectangle?

1. x2−10 because the area expression can be rewritten as (x2−10)(x2+10) which equals (x2−10)((x2−10)+20).

2. x2+10 because the area expression can be rewritten as (x2+10)(x2−10) which equals (x2+10)((x2+10)−20).

3. x2−30 because the area expression can be rewritten as (x2+10)(x2−10) which equals (x2+10)((x2−30)+20).

4. x2+30 because the area expression can be rewritten as (x2−10)(x2+10) which equals (x2−10)((x2+30)−20).

Answer 1

x2−10 because the area expression can be rewritten as (x2−10)(x2+10) which equals (x2−10)((x2−10)+20).

Explanation:

The area of the rectangle is
`x^(4) -100`, which can be factored as
`(x^(2) +10)(x^(2) -10)`, because it's the difference of two perfect squares.

But, we know that
`l=20+w`, where
`l` is the length and
`w` is the width.

Additionally,
`l=x^(2) -10+20` and
`w=x^(2) -10`, which means
`l=x^(2) +10`.

Therefore, the right answer is A.

Answer 2

`1. x^2-10` because the area expression can be rewritten as
`(x^2-10)(x^2+10)`which equals
`(x^2-10)((x^2-10)+20).`

Explanation:

Area of the rectangle
`=(x^4-100)`

`x^4-100=(x^2)^2-10^2\\$Applying difference of two squares: a^2-b^2=(a-b)(a+b)\\(x^2)^2-10^2=(x^2-10)(x^2+10)`

Since the length of a rectangle is 20 units more than its width.

`Width: x^2-10\\Length=x^2+10=x^2-10+20`

The correct option is therefore 1.