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How is adding and subtracting rational expressions similar to adding and subtracting numerical fractions?
See solution.
We are asked to explain how to add and subtract rational expressions. Recall that a rational expression is a fraction whose numerator and denominator are polynomials. Rational Expression: polynomial/polynomial When adding and subtracting rational expressions we can use the same rules as when adding and subtracting numerical fractions. Therefore, we can add and subtract rational expressions only if they have the same denominator. If the expressions have different denominators, we can rewrite them using their least common denominator. Let's review these two cases!
If rational expressions have the same denominator, we add them by adding their numerators.
Sum | Add Numerators | Simplify Numerator |
---|---|---|
1/x^2+3/x^2 | 1+ 3/x^2 | 4/x^2 |
y+1/2y+5+2y-1/2y+5 | y+1+ 2y-1/2y+5 | 3y/2y+5 |
z^2+3/z+1+z-1/z+1 | z^2+3+ z-1/z+1 | z^2+z+2/z+1 |
Likewise, if rational expressions have the same denominator, we subtract them by subtracting their numerators. When the numerator which is being subtracted has more than one term, we have to put parentheses around it. Then we can use the Distributive Property to further simplify our result.
Difference | Subtract Numerators | Distributive Property | Simplify Numerator |
---|---|---|---|
1/x^2-3/x^2 | 1- 3/x^2 | Does not apply | - 2/x^2 |
y+1/2y+5-2y-1/2y+5 | y+1-( 2y-1)/2y+5 | y+1-2y+1/2y+5 | 2-y/2y+5 |
z^2+3/z+1-z-1/z+1 | z^2+3-( z-1)/z+1 | z^2+3-z+1/z+1 | z^2-z+4/z+1 |
If the rational expressions that are being added or subtracted have different denominators, we can take the following steps.
Sum or Difference | LCD | Rewritten Expressions | Result |
---|---|---|---|
2/y+y/y+1 | y(y+1)=y^2+y | 2y+2/y^2+y+y^2/y^2+y | y^2+2y+2/y^2+y |
6/5x^2-1/2x | 5* x* x* 2=10x^2 | 12/10x^2-5x/10x^2 | 12-5x/10x^2 |
x^2/x^2-1-x/x+1 | (x-1)(x+1)=x^2-1 | x^2/x^2-1-x^2-x/x^2-1 | x/x^2-1 |