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What is a rational function in algebraic terms?
Do you need to find a common denominator to be able to add or subtract rational functions? Is there a step-by-step process to use in the addition or subtraction of rational functions?
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M$1 Answer
A rational function is a function that can be expressed as the ratio of two polynomial functions.
For example. If you have f(x) = g(x)/h(x) where g(x) = x^2+4x+4 and h(x) = x^2-9 and d(x) = a(x)/b(x) where a(x) = x^2+6x+9 and b/(x) = (x+2) and you want f(x) - d(x) then the problem is set up as follows
{(x^2+4x+4)}/(x^2-9) - {(x^2+6x+9)}/(x+2)
to subtract these we need a common denominator, but first factor everything that you can
(x^2+4x+4) = (x+2)(x+2)
(x^2-9) = (x+3)(x-3)
(x^2+6x+9) = (x+3)(x+3)
The problem now looks like this:
(x+2)(x+2)/(x+3)(x-3) - (x+3)(x+3)/(x+2)
the common denominator is (x+3)(x-3)(x+2)
Multiply the (x+2)(x+2)/(x+3)(x-3) by (x+2)/(x+2) and the (x+3)(x+3)/(x+2) by (x+3)(x-3)/(x+3)(x-3) to get
(x+2)(x+2)(x+2)/(x+3)(x-3)(x+2) - (x+3)(x-3)(x+2)/(x+3)(x-3)(x+2)
Now there is a common denominator so you can subtract to get
{(x+2)^3-(x+3)(x-3)(x+2)}/(x+3)(x-3)(x+2)
For example. If you have f(x) = g(x)/h(x) where g(x) = x^2+4x+4 and h(x) = x^2-9 and d(x) = a(x)/b(x) where a(x) = x^2+6x+9 and b/(x) = (x+2) and you want f(x) - d(x) then the problem is set up as follows
{(x^2+4x+4)}/(x^2-9) - {(x^2+6x+9)}/(x+2)
to subtract these we need a common denominator, but first factor everything that you can
(x^2+4x+4) = (x+2)(x+2)
(x^2-9) = (x+3)(x-3)
(x^2+6x+9) = (x+3)(x+3)
The problem now looks like this:
(x+2)(x+2)/(x+3)(x-3) - (x+3)(x+3)/(x+2)
the common denominator is (x+3)(x-3)(x+2)
Multiply the (x+2)(x+2)/(x+3)(x-3) by (x+2)/(x+2) and the (x+3)(x+3)/(x+2) by (x+3)(x-3)/(x+3)(x-3) to get
(x+2)(x+2)(x+2)/(x+3)(x-3)(x+2) - (x+3)(x-3)(x+2)/(x+3)(x-3)(x+2)
Now there is a common denominator so you can subtract to get
{(x+2)^3-(x+3)(x-3)(x+2)}/(x+3)(x-3)(x+2)
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