SOLUTION: Can someone please help me with this problem: Perform the indicated operations. Give the answer in lowest terms. 16x / 7(4x+1) - 1 / 7x(4x+1) + 3 / x

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Can someone please help me with this problem: Perform the indicated operations. Give the answer in lowest terms. 16x / 7(4x+1) - 1 / 7x(4x+1) + 3 / x      Log On


   

Question 201126: Can someone please help me with this problem:
Perform the indicated operations. Give the answer in lowest terms.
16x / 7(4x+1) - 1 / 7x(4x+1) + 3 / x
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
As you probably know, in order to ad or subtract fractions, the denominators must be the same. The first difficulty with this problem is figuring out how to get the denominators the same.
Let's look at the denominators:
7(4x+1) = 7 * (4x+1)
7x(4x+1) = 7 * x * (4x+1)
x

If you look at the factors of the denominators you will that there is a total of 3 different factors spread across the three denominators: 7, x and (4x+1). The lowest common denominator (LCD) will be the product of these three factors: 7 * x * (4x+1).
To change a fraction so it has the LCD, multiply the numerator and the denominator by the factors of the LCD that it does not already have in its denominator:
16x%2F%287%284x%2B1%29%29already has factors of 7 and (4x+1) in its denominator. The only missing factor is x. So multiply the numerator and denominator of 16x%2F%287%284x%2B1%29%29 by x%2Fx.
1%2F%287x%284x%2B1%29%29 is not missing any of the factors of the LCD in its denominator. It is "ready to go".
3%2Fx only has a factor of x in its denominator. So we need to multiply the numerator and denominator by both missing factors: 7 and (4x+1).
Putting this all together:
16x%2F%287%284x%2B1%29%29+-+1%2F%287x%284x%2B1%29%29+%2B+3%2Fx

Multiply out the numerators (but leave the denominators factored):

Now we can add and subtract (combining only like terms, of course):
%2816x%5E2%2B84x%2B20%29%2F%287x%284x%2B1%29%29
Now we try to reduce the fraction by canceling common factors. In order to do this we need the numerator factored. (The denominator should still be factored.)
Factor out the Greatest Common Factor (GCF) which is 4:
%284%284x%5E2%2B21x%2B5%29%29%2F%287x%284x%2B1%29%29
Now factor the trinomial: 4x%5E2%2B21x%2B5%29
%284%284x%2B1%29%28x%2B5%29%29%2F%287x%284x%2B1%29%29
We can now see that (4x+1) is a factor of both the numerator and the denominator. They will cancel giving:
%284%28x%2B5%29%29%2F%287x%29%29
This may be an acceptable answer. Or you could multiply out the numerator
%284x%2B20%29%2F%287x%29
and then separate it into 2 fractions ("un-adding" them)
%284x%29%2F%287x%29%2B20%2F%287x%29
In the first fraction the x's cancel leaving
4%2F7%2B20%2F%287x%29

Your answer may be any of the following depending on which is considered "lowest terms":
%284%28x%2B5%29%29%2F%287x%29%29
%284x%2B20%29%2F%287x%29
4%2F7%2B20%2F%287x%29