Final answer:
To simplify the given rational expression, factor the denominators, find the common denominator, combine the expressions, and simplify by combining like terms and reducing. The expression simplifies to (r^2 - 4r + 5) / (r - 4) - (r^2 / 2).
Step-by-step explanation:
The student's question involves simplifying a complex rational expression:
r^2 - 4r + 5 / (r - 4) - r^2 ÷ (2r - 8) / (r - 4)
To simplify this expression, we have to follow several steps:
Factor the denominators if possible.
Find a common denominator.
Combine the rational expressions.
Simplify the result by combining like terms and reducing.
The denominators (r - 4) and (2r - 8), can be simplified since 2r - 8 is 2(r - 4), hence they have a common factor.
Since division by a fraction is equivalent to multiplication by its reciprocal, the expression becomes:
(r^2 - 4r + 5) / (r - 4) - (r^2) / (2(r - 4)) × (r - 4)
After multiplying the second term by the reciprocal, we combine like terms:
(r^2 - 4r + 5) / (r - 4) - (r^2 / 2)
Simplifying further, the rational expression is reduced.