Question

1. the length of a rectangle is 20 units more than its width. the area of the rectangle is ( x^4 - 100 ).

which statement about the width of the rectangle is true?
a. ( 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) )
b. ( 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) )
c. ( x^2 + 30 ) because the area expression can be rewritten as ( (x^2 - 10)(x^2 + 10) ) which equals ( (x^2 - 10)((x^2 + 30) - 20) )
d. ( x^2 - 30 ) because the area expression can be rewritten as ( (x^2 + 10)(x^2 - 10) ) which equals ( (x^2 + 10)((x^2 - 30) + 20) )

Explanation:

Step1: Factor the area expression

Use difference of squares: $a^2-b^2=(a+b)(a-b)$.
$x^4 - 100 = (x^2)^2 - 10^2 = (x^2+10)(x^2-10)$

Step2: Match to rectangle dimensions

Length = width + 20. Check which factor pair fits:
If width = $x^2-10$, then length = $(x^2-10)+20 = x^2+10$, which matches the factored area.

Answer:

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