Question

Help in further simplifying the denominator is solving rational expressions!!! MEDAL!!!!!!

Answer 1

\[\large 6v - \frac{ v-5 }{ 16v^2 - 108v + 72 }\]

Answer 2

@mathmale

Answer 3

I have the numerator right, but the denominator shows it more simplified. How do I make the denominator more simplified?

Answer 4

Hello! What's the object of this problem? Are you to combine terms using the LCD?
Regarding simplifying the LCD, what's the largest common factor?

Answer 5

My original answer:
\[\large \frac{ 96v^3 - 648v^2 + 431v + 5 }{ 16v^2 - 108v + 72 }\]

Answer 6

The answer key says:
\[\large 4(4v - 3)(v - 6)\]
for the denominator.

Answer 7

Look carefully at the denom. I see that it has a common factor. Unfortunately, this common factor is not also a factor of the numerator. Not much more you can do to simplify this algebraic fraction other than to identify the greatest common factor of the den, factor it out of the den. and then factor what's left of the den. Can you do these things?

Answer 8

So the greatest common factor would be 4?

Answer 9

That's right. Factor out the 4 and then concentrate on what's left of the den. Can you now factor that "what's left"?

Answer 10

Okay so I would have:
\(\large 4(4v^2 - 27v + 18)\)

Answer 11

@mathmale

Answer 12

@Michele_Laino

Answer 13

Please demo how you'd go about factoring (4v^2 - 27v + 18).

Answer 14

Yes, 4 is the common factor of the den. Once you've factored that out, you need to factor the trinomial which is left over.

Answer 15

What methods do you know for doing that?

Answer 16

Would it be possible to complete the square? Or is there an easier way?

Answer 17

Certainly completing the square would work, but there are easier ways in this particular situation.

Answer 18

Can you please show me?

Answer 19

Look carefully at 4v^2 - 27v + 18.
One way to start would be to assume that both the two binomial factors begin with 2v.

Answer 20

I don't understand :(

Answer 21

A more sophisticated method would be to multiply the coeff. of the first and last terms together and then list all the possible factors of the product. 4 times 18 is 72.

What are factors of 72? 1 and 72, 2 and 36, 3 and 24, and so on.

In your shoes I'd try writing 2 binomials that begin with 2x: (2x + ?? ) (2x + ?? )
and choose from the various factors of 18 to find the 2nd term within each factor.

Answer 22

And how would we know that a certain pair of factors are correct?

Answer 23

For example:
(2v + 3)(2v+6): both factors begin with 2v, and 3 and 6 are factors of 18.
You'll see right away if you multiply these together that they don 't give you 4v^2 - 27v + 18.

Answer 24

The factors are correct if multiplying them together gives y ou 4v^2 - 27v + 18.

Answer 25

So is this like trial and error?

Answer 26

One more suggestion: synthetic division. Are you familiar with that?

Answer 27

No but can you show me?

Answer 28

Most people have to start with trial and error, and then learn more sophisticated approaches.

Answer 29

I'd be happy to show you synth. div., but would prefer to solve this problem first using methods you already know.

Answer 30

So, of the methods we've already discussed, which are most familiar to you?

Answer 31

So another thing that I have learned was to multiply the \(a\) and the \(c\) terms. I then list out the factors of that product and then list the factors of that product. Is that fine?

Answer 32

A few min. ago I suggested starting with (2v )(2v ), simply because 2v * 2v = 4v^2.
These 2's may or may not be correct for the polynomial you're trying to factor.
I've also suggested that you multiply the first and last terms together

Answer 33

obtaining 4*18 = 72. Then, you might try (x )(4x ) or (2x )(2x )

You'd then need to try pairs of factors of 18 to find the 2nd terms of each of these factors.
Again there are easier ways to do this problem, but I'd prefer you use the most basic approach until you're confident about being able to use that....then move on up.

Answer 34

How would you factor x^2 - 5x + 6? Factoring this might help refresh your memory.

Answer 35

You told me you already have the answer and have shared it. You could try dividing one of those factors in to 4v^2 - 27v + 18 to find the other factor.

Want to continue with this discussion, or put it on hold and return to it later? Your choice.

Answer 36

I will put it on hold while I am trying to solve. Is that fine?

Answer 37

Yes, of course. And of course this kind of problem becomes easier with practice and experience. Good luck!

Answer 38

I got it!!!

Answer 39

Thank you @mathmale