Solved Agebraic Fraction Exercises

a

 

We extract the common factor from the numerator and denominator expressions, thus we have

 

 

Now, we "cancel said common factor", thus our simplification becomes

 

 

 

b

We extract the common factor from the numerator

 

 

 

We multiply numerator and denominator by , so we will obtain an equivalent fraction

 

 

 

Distributing the sign in the denominator we have

 

 

 

Canceling the common factor in the denominator and numerator we obtain

 

 

 

c

 

We apply the remainder theorem:

 

 

 

 

We divide by Ruffini both the numerator and denominator expressions

 

 

 

 

We have an exact division, thus and therefore

 

 

We simplify by canceling the common factor from the numerator and denominator

 

 

 

Note that the denominator , however, neither of these factors is in the numerator, so it cannot be canceled or simplified further in that sense, but we can write the expression as

 

 

 

Either of the two expressions in the equality are correct and valid.

 

 

d

 

Using the quadratic formula we obtain the roots of the numerator polynomial and the denominator polynomial, this will help us express said polynomials as multiplication of binomials defined by their roots

 

 

 

 

We factor:

 

 

We simplify

 

 

e

 

Using the quadratic formula we obtain the roots of the numerator polynomial and the denominator polynomial, this will help us express said polynomials as multiplication of binomials defined by their roots

 

 

 

 

We factor:

 

 

We simplify

 

 

f

 

In the numerator we use the remainder theorem and Ruffini's rule to find the integer roots

 

 

The divisors of are: {}

 

 

We divide by Ruffini

 

 

The numerator satisfies

 

 

 

We can continue factoring the trinomial in the same way or using the quadratic formula

 

 

 

 

In the denominator we extract the common factor

 

 

 

To factor the trinomial we use the general formula

 

 

 

 

Thus, our initial expression can be written as

 

 

 

We simplify

 

We need to find the common denominator, for this we have to find the LCM of the denominators, note that

 

We divide the common denominator by the denominators of the given fractions and multiply the result by the corresponding numerator

 

We need to find the common denominator, for this we have to find the LCM of the denominators, note that

 

We divide the common denominator by the denominators of the given fractions and multiply the result by the corresponding numerator

We need to find the common denominator, for this we have to find the LCM of the denominators, note that

 

We divide the common denominator by the denominators of the given fractions and multiply the result by the corresponding numerator

We need to find the common denominator, for this we have to find the LCM of the denominators, note that

 

 

Thus

 

 

We divide the common denominator by the denominators of the given fractions and multiply the result by the corresponding numerator

 

 

We extract the common factor

 

 

We simplify

 

We need to find a common denominator, for this we have to find the LCM of the denominators. Note that

 

 

Thus

 

 

We divide the common denominator by the denominators of the given fractions and multiply the result by the corresponding numerator and operate

 

 

 

Furthermore, we have that , thus we obtain
 

 

 

We simplify

 

We have a sum times difference which we express as a difference of squares, therefore

 

 

We find a common denominator
 

 

 

 

We operate

 

We have a sum times difference which we express as a difference of squares, therefore

We find a common denominator
 

 

We extract the common factor and operate
 

We multiply

The division of two algebraic fractions is another algebraic fraction whose numerator is the product of the numerator of the first times the denominator of the second, and as denominator the product of the denominator of the first times the numerator of the second.

 

 

The second binomial is a sum of cubes:

 

The trinomial in the denominator is a perfect square trinomial and the binomial is a difference of squares that factors as a sum times difference.

 

 

 

We simplify

 

 

or

 

By dividing we have

 

 

The first factor decomposes using the remainder theorem and division by Ruffini.

In the second factor we extract the common factor , we are left with a perfect square trinomial which we express as a binomial squared.

The first factor of the denominator is a second-degree trinomial that factors using the general formula. In the second factor we extract the common factor . Thus, our original expression would be

 

simplifying a bit

 

 

We multiply the numerator and denominator by , obtaining an equivalent fraction.

 

 

 

We simplify

 

The division of two algebraic fractions is another algebraic fraction whose numerator is the product of the numerator of the first times the denominator of the second, and as denominator the product of the denominator of the first times the numerator of the second.

The second binomial in the denominator is a difference of cubes:

The first trinomial in the numerator is a perfect square trinomial.

We simplify

We find a common denominator

 

 

 

The division of two algebraic fractions is another algebraic fraction whose numerator is the product of the numerator of the first times the denominator of the second, and as denominator the product of the denominator of the first times the numerator of the second.

 

 

 

We simplify

 

We find a common denominator

 

The division of two algebraic fractions is another algebraic fraction whose numerator is the product of the numerator of the first times the denominator of the second, and as denominator the product of the denominator of the first times the numerator of the second.

 

We simplify

First we subtract and then we take the inverse of the result.

 

First we add and take the inverse of the result, then we add again and so on until we find our result.