Thus, We subtract fractions with unlike denominators in a similar way that we add such fractions. We build each fraction to an equivalent fraction with this denominator to get Now, adding numerators yields Again, special care must be taken with binomial numerators.
Thus, We subtract fractions with unlike denominators in a similar way that we add such fractions. We build each fraction to an equivalent fraction with this denominator to get Now, adding numerators yields Again, special care must be taken with binomial numerators.However, we first write each fraction in standard form. Example 2 Write the difference of as a single term.The product of two fractions is defined as follows.
One case is In general, Example 3 When the fractions in a quotient involve algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying.
Example 4 The sum of two or more arithmetic or algebraic fractions is defined as follows: The sum of two or more fractions with common denominators is a fraction with the same denominator and a numerator equal to the sum of the numerators of the original fractions.
To find the LCD: Example 1 Find the lowest common denominator of the fractions Solution The lowest common denominator for contains among its factors the factors of 12, 10, and 6. (This number is the smallest natural number that is divisible by 12, 10, and 6.) The LCD of a set of algebraic fractions is the simplest algebraic expression that is a multiple of each of the denominators in the set.
Thus, the LCD of the fractions because this is the simplest expression that is a multiple of each of the denominators.
This is precisely the same notion as that of dividing one integer by another; a ÷ b is a number q, the quotient, such that bq = a. To solve this equation for q, we multiply each member of the equation by .
Thus, In the above example, we call the number the reciprocal of the number .
Example 2 Find the product of Solution First, we divide the numerator and denominator by the common factors to get Now, multiplying the remaining factors of the numerators and denominators yields If a negative sign is attached to any of the factors, it is advisable to proceed as if all the factors were positive and then attach the appropriate sign to the result.
A positive sign is attached if there are no negative signs or an even number of negative signs on the factors; a negative sign is attached if there is an odd number of negative signs on the factors.
In general, The quotient of two fractions equals the product of the dividend and the reciprocal of the divisor.
That is, to divide one fraction by another, we invert the divisor and multiply.