Operations with Algebraic Fractions.

10.15 Calculate:

Answer:

Solution:

The l.c.m. of the denominators is the product of both:

Divide it by each denominator. Multiply their quotient by their corresponding numerator. This is an example of the first fraction:

We have as quotient. We multiply it by the first numerator, which is also . We follow the same process with the second fraction:

The first numerator is the square of the difference of two numbers and the LCD is equal to a difference of squares:

We develop the squares of the difference and the sum of the squares of two numbers.

After the negative sign joining the fractions, we write a parenthesis. When we remove them, we change the sign of each term. Then, we reduce any similar terms:

10.16 Calculate:

Answer:

or:

Solution:

The first denominator is the result of multiplying the sum of two numbers by their difference; . Then, the l.c.m. of the denominators will be .

We divide by the second denominator :

Be careful when you remove a parenthesis with a negative sign in front. We simplify any similar terms:

Since a negative sign in front of a fraction only affects the numerator, we could write this result:

10.17 Calculate:

Answer:

or:

Solution

Step-by-step:

10.18 Calculate:

Answer: or :

Solution:

The third denominator: is not a product of the other two denominators. The second denominator won't work; it would have to be . The first denominator works because we can write it as.

We have already mentioned that a negative sign in front of a fraction only affects the numerator and this is important in what you are about to see. If we change the sign to both terms in a fraction (numerator and denominator), the result is the same:

Pay attention to the next example:

If we change the signs to the numerator and denominator, the result will be the same:

Notice now the same example with the negative sign in front of the fraction:

As you can see, the result of all cases is the same.

If we have, it would be the same as writing: because the negative sign in front of a fraction only affects the numerator.

Imagine that instead of we want the denominator to be: As you can see, we changed the signs of each term in the denominator. If we change the signs of the terms in the denominator, __we also have to change the signs in the numerator__:

Since the sign in front of a fraction only affects the numerator, we can write:

With this in mind, we can solve the exercise:

We can also solve it by changing the signs of the numerator and denominator in the second fraction: , which is the same as .

At first look, the results are: .

They seem different but in fact they are the same:

Imagine that x = 2. Substitute x with 2 in the first answer:

Substitute x with 2 in the second answer:

We have the same result in both cases.

If we change the signs of each term in the fraction:

the result doesn't vary.

10.19 Calculate:

Answer: .

Solution:

If the denominator of the third fraction were , it would be the l.c.m. of the denominators. This demands a change in the signs of each term in the numerator and denominator in the exercise:

10.20 Calculate:

Answer: Any of the three answers is valid:

Solution:

We need to analyse each fraction before calculating. In the denominator of the first fraction, we can take 10 as a common factor. In the denominator of the third fraction, we can take 4 as a common factor. In both cases, we can simplify with a factor of its respective numerator:

From this, we see that the l.c.m. of the denominators is .