simplify the following polynomial expressions and state restrictions.
a)3/x^2-7x+10-5x/x^2-4
b)6x^2-54y^2/x^2+4xy-21y^2
a) To simplify the expression 3/x^2-7x+10-5x/x^2-4, we first need to factor the denominators.
The denominator of the first fraction can be factored as (x-5)(x-2).
The denominator of the second fraction can be factored as (x+2)(x-2).
Now our expression becomes: 3/(x-5)(x-2) - 5x/(x+2)(x-2)
Next, we need to find a common denominator for both fractions. The common denominator is (x-5)(x-2)(x+2).
For the first fraction, we need to multiply the numerator and denominator by (x+2) to get a common denominator.
3(x+2)/[(x-5)(x-2)(x+2)] = (3x+6)/[(x-5)(x-2)(x+2)]
For the second fraction, we need to multiply the numerator and denominator by (x-5) to get a common denominator.
-5x(x-5)/[(x-5)(x-2)(x+2)] = -5x(x-5)/[(x-5)(x-2)(x+2)] = -5x^2 + 25x / [(x-5)(x-2)(x+2)]
Combining the fractions, our simplified expression is:
(3x+6)/[(x-5)(x-2)(x+2)] - (5x^2 - 25x)/[(x-5)(x-2)(x+2)]
b) To simplify the expression 6x^2-54y^2/x^2+4xy-21y^2, we cannot simplify it any further because there are no common factors that can be cancelled out. The restriction for this expression is that x cannot be equal to 0 to avoid division by zero.