Rational expressions (§§7.1&7.2)
A rational expression
is made of two polynomial expressions:
the numerator (or top)
and the denominator (or bottom),
where the denominator is not the constant polynomial 0.
A rational expression is simplified
if - the top and bottom have only integer coefficients,
- the leading coefficient on the bottom is positive,
- the coefficients have no common integer factor greater than 1, and
- the top and bottom have
no common polynomial factor that is not constant.
We can simplify rational expressions by factoring the top and bottom
and cancelling any common factors;
we may then leave the expression in factored form.
(If the leading coefficient in the numerator is negative
and we leave the expression in factored form,
then we place the minus sign in front of the entire expression.)
We add, subtract, multiply, and divide rational expressions
using the same techniques as for rational numbers.
But systematic factoring is now more important,
since less can be done by trial and error.
When multiplying or dividing,
you should factor all of the numerators and denominators,
and there is no need to get common denominators;
when adding or subtracting, you only need to factor the denominators at first,
and you need to make all of these denominators the same.
(You should still factor the numerator of the final answer
when adding or subtracting,
but that's only to simplify the result.)
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