Answer:
x
Step-by-step explanation:
We are required to simplify the rational expression:
[tex]\dfrac{4}{x^2+4x-5} -\dfrac{5}{x^2-25}[/tex]
First, we factorize the denominators where possible
[tex]x^2+4x-5=x^2+5x-x-5=x(x+5)-1(x+5)=(x-1)(x+5)\\x^2-25=x^2-5^2 \\$Applying difference of two squares\\x^2-5^2=(x-5)(x+5)[/tex]
Therefore, our rational expression becomes:
[tex]\dfrac{4}{(x-1)(x+5)} -\dfrac{5}{(x-5)(x+5)}\\$Taking LCM\\=\dfrac{4(x-5)-5(x-1)}{(x-1)(x-5)(x+5)}\\=\dfrac{4x-20-5x+5}{(x-1)(x-5)(x+5)}\\=\dfrac{-x-15}{(x-1)(x-5)(x+5)}\\$Factoring out minus in the numerator and -(x-5)=(5-x)\ in the denominator\\=\dfrac{x+15}{(x-1)(5-x)(x+5)}[/tex]
Therefore, the leading coefficient of the numerator is x.
