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p(x)/q(x)
Here, p(x) and q(x) are polynomials and q(x)≠ 0. The expression below is an example of a rational expression. x^2-7/x^3+5x A rational expression is said to be written in its simplest form if the numerator and denominator have no common factors.
| Rational Expressions | |
|---|---|
| Not in Simplest Form | In Simplest Form |
| xy/x(x-3)(y+2) | x-1/x+1 |
| x^4+x^2/x^2+1 | x^3+7/x^2-x-6 |
Notice that for some of the expressions shown in the table, there are some x-values that make the denominator 0. For example, the denominator of x-1x+1 is 0 when x=- 1. Any value of a variable for which a rational expression is undefined is called an excluded value.
| Expression | Restriction | Excluded Value(s) |
|---|---|---|
| x-1/x+1 | x+1≠ 0 | x ≠ - 1 |
| x^3+7/x^2-x-6 | x^2-x-6≠ 0 | x≠ - 2 and x≠ 3 |
| xy/x(x-3)(y+2) | x(x-3)(y+2)≠ 0 | x≠0, x ≠ 3, and y ≠ - 2 |
| x^4+x^2/x^2+1 | There is no real number that makes x^2+1 zero | None |
Simplifying a rational expression can remove some of the excluded values that appear in the original expression. A rational expression and its simplified form must have the same domain in order for them to be equivalent expressions. This means that the excluded values that are no longer visible in the simplified expression must still be declared.
| Equivalent Expressions | |
|---|---|
| Rational Expression | Simplified Form |
| x-3/(x+2)(x-3), x≠ - 2, 3 | 1/x+2,x≠ - 2, 3 |
| x^2+2x+1/x^2-1, x≠ - 1, 1 | x+1/x-1,x≠ - 1,1 |
| x^3-2x^2+x/x^2,x≠ 0 | x^2-2x+1/x, x≠ 0 |