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When adding and subtracting rational expressions, the same rules apply as when adding and subtracting fractions.
If the rational expressions have a common denominator, the numerators can be added or subtracted directly.
P(x)/Q(x) ± R(x)/Q(x)=P(x) ± R(x)/Q(x)
Here, P(x), Q(x), and R(x) are polynomials and Q(x)≠ 0.
If the denominators are different, the expressions have to be manipulated to find a common denominator before they can be added or subtracted. One way of doing this is to multiply both numerator and denominator of one of the rational expressions with the denominator of the other, and vice versa.
P(x)/Q(x) ± R(x)/S(x)=P(x) S(x) ± R(x)Q(x)/Q(x)S(x)
Here, P(x), Q(x), R(x), and S(x) are polynomials, Q(x)≠ 0, and S(x)≠ 0. Another way is to find the least common multiple of the denominators, or the least common denominator. Consider adding the following rational expressions to put theory into practice. x+2/2x-2 + 2x+1/x^2-4x+3 The result can be found in four steps.
| Denominator | Factored Form |
|---|---|
| 2x-2 | 2(x-1) |
| x^2-4x+3 | (x-1)(x-3) |
The least common denominator is the product of the highest power of each prime factor. LCD: 2(x-1)(x-3)
a/b=a * (x-3)/b * (x-3)
Multiply parentheses
Add terms
a/b=a * 2/b * 2
Distribute 2
Multiply
The result is in simplest form. Note that the excluded values for the sum are 1 and 3. x+4/2(x-3), x ≠ 1 The restrictions on the domain of a sum or difference of rational expressions consist of the restrictions to the domains of each expression.
Note that the process for subtracting rational expressions is similar.