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## Rational Expression and Equations

Please help me out

Add or subtract this rational expressions.

(1)

10                7
--------------- + ------------
x^2-16x+60       x^2-100

(2)

3                   2
.................... -  ...............
x^2-6x-16x        x^2-2x-48
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# 1 Answer

To simplify rational expressions, it is often helpful to factor all of the involved polynomials:

(1)

10/(x^2-16x+60) + 7/(x^2-100)

= 10 /[(x-6)(x-10)] + 7/[(x-10)(x+10)]

Since both terms in the expression have an (x-10) in the denominators, we only need an extra (x-6) in the first term and an (x+10) in the second.  Now, we cannot willy-nilly add terms to products, but we can multiply expressions by 1 and preserve them.  To wit:

10(x+10) /[(x-6)(x-10)(x+10)] + 7(x-6)/[(x-10)(x+10)(x-6)]

= (10x+100)/[~] + (7x-42)/[~]    (I am using ~ for the common denominator.)
= (10x+100 + 7x-42)/[~]    (We can add two fractions since we have a common denominator.)
= (17x+58)/[(x-6)(x-10)(x+10)]

For your second expression, please verify the first denominator as I presume you added an x in there.