The numerator is the upstairs parts of a fraction. The denominator is the downstairs part of the fraction. IF there is a plus or minus sign between fractions, each is a seperate term. You do NOT try to cross out factors in one term and jump across the plus or minus sign and cross out factors in a different term. The correct math is to get a common denominator and combine them.
Steps to Combining Fractions:
The initial problem
Step 1. Add parentheses to any terms(seperated by a plus or minus) in the numerator and denominator so they are factors.
Step 2. Multiply the numerator and denominator of each term by any factors that the terms have in their denominators that are missing. This will give the terms a common denominator.
Step 3. Once the terms have common (matching) denominators, combine by writing the common denominator and doing the math with the numerators.
Step 4. Distribute or FOIL any parts that are trapped in parentheses.
Step 5. Do the math.
Step 6. Combine Like Terms. If the part behind the coefficient matches (is an x, for example), combine by doing the math with the coefficients and writing the variable behind it.
Step 7. See if the numerator can be simplified by finding the greatest common factor. There is a common factor in all 3 terms(a 3), so write it out front and divide it out of each term.
This technique only works with factors. FACTORS DO NOT HAVE A PLUS OR MINUS SIGN BETWEEN THEM AND ANOTHER FACTOR. The rule is: If there is the same factor in the numerator and denominator it will simplify to a 1, so cross them out.
Step1. DO NOT CROSS OUT ANYTHING. There are no factors here. Everything is terms (plus or minus is seperating the pieces).
Step 2. FOIL the numerator backwards to turn it into linear factors. The factors are x+5 and x-3.
Step 3. FOIL the denominator backwards to turn it into linear factors. The pattern here is the difference of two perfect squares. The factors are x+3 and x-3.
Step 4. See if there are any factors that match in the numerator and denominator. Here, both have an x-3 factor so that can be crossed out.
Done! Do not cross out the x's. They are not factors, they are terms.
You can't leave it there in a final answer. Multiply the numerator and denominator by the radical in the denominator. That will make the denominator just the number and move the radical to the numerator.
Dividing Fractions:
You need to know how to make the reciprocal of a fraction. The numerator and denominator have to switch positions.
Once you can make the reciprocal of the denominator, you divide a fraction by rewriting it as the fraction in the numerator multiplied by the reciprocal of the fraction in the denominator.
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