Rational functions: combine and simplify terms

[cptstubing]

Homework Statement

(4a/a+4)+(a+2/2a)

Homework Equations

Just combine and then factor out

The Attempt at a Solution

It's actually fairly simple, but I'm having difficulty at the end.
/multiply each term by opposite denominator
4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4)
/combine
4a(2a)+(a+2)(a+4) / a+4(2a)

=9a²+6a+8 / a+4(2a) ---- do I try to factor the numerator?
=a²+6a+72 / a+4(2a) ---- to this and keep going? There are no real roots.

=3a(3a+2)+8 / a+4(2a) ---- or do I factor the 9a²+6a, or possibly the 6a+8 and leave the 9a²?

I'd really appreciate some help with this one.

 

[PeroK]

Homework Statement

(4a/a+4)+(a+2/2a)

Homework Equations

Just combine and then factor out

The Attempt at a Solution

It's actually fairly simple, but I'm having difficulty at the end.
/multiply each term by opposite denominator
4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4)
/combine
4a(2a)+(a+2)(a+4) / a+4(2a)

=9a²+6a+8 / a+4(2a) ---- do I try to factor the numerator?
=a²+6a+72 / a+4(2a) ---- to this and keep going? There are no real roots.

=3a(3a+2)+8 / a+4(2a) ---- or do I factor the 9a²+6a, or possibly the 6a+8 and leave the 9a²?

I'd really appreciate some help with this one.

What are you trying to do "at the end" exactly?

 

[cptstubing]

What are you trying to do "at the end" exactly?

The homework question is to "combine and simplify", and state restrictions. X cannot equal such and such a number.
I should be able to cancel some factors in the questions, but I cannot seem to.

 

[Mark44]

Homework Statement

(4a/a+4)+(a+2/2a)

You have used parentheses, but not in a way that makes sense.
Using the usual rules of operator precedence, this is what you wrote:
$(\frac {4a}{a} + 4) + (a + \frac 2 2 a)$
The above simplifies to this
(4 + 4) + (a + a) = 8 + 2a

Presumably this isn't what you meant, which means you need to place parentheses around each numerator or denominator that has more than one term or factor, NOT just around the entire rational expression.

I'm guessing that what you really meant was this:
$\frac{4a}{a + 4} + \frac{a + 2}{2a}$
Using LaTeX, no parentheses are needed, but if you write the expression without LaTeX, it should probably be like so:
4a/(a + 4) + (a + 2)/(2a)
or possibly this:
4a/(a + 4) + (a + 2)a/2

Please tell us the expression you're starting with.

Homework Equations

Just combine and then factor out

The Attempt at a Solution

It's actually fairly simple, but I'm having difficulty at the end.
/multiply each term by opposite denominator
4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4)
/combine
4a(2a)+(a+2)(a+4) / a+4(2a)

=9a²+6a+8 / a+4(2a) ---- do I try to factor the numerator?
=a²+6a+72 / a+4(2a) ---- to this and keep going? There are no real roots.

=3a(3a+2)+8 / a+4(2a) ---- or do I factor the 9a²+6a, or possibly the 6a+8 and leave the 9a²?

I'd really appreciate some help with this one.

 

[PeroK]

The homework question is to "combine and simplify", and state restrictions. X cannot equal such and such a number.
I should be able to cancel some factors in the questions, but I cannot seem to.

You noted that the quadratic in the numerator does not factor, so I'm not sure what sort of simplification you think is possible. You've got to know when to stop!

 

[Mark44]

=9a²+6a+8 / a+4(2a) ---- do I try to factor the numerator?
=a²+6a+72 / a+4(2a) ---- to this and keep going? There are no real roots.

=3a(3a+2)+8 / a+4(2a) ---- or do I factor the 9a²+6a, or possibly the 6a+8 and leave the 9a²?

I'd really appreciate some help with this one.

You could start by writing your rational expressions properly. None of the expressions you wrote means what you think it does. When a numerator or denominator has more than one term, you need to surround it with parentheses.

 

[cptstubing]

You have used parentheses, but not in a way that makes sense.
Using the usual rules of operator precedence, this is what you wrote:
$(\frac {4a}{a} + 4) + (a + \frac 2 2 a)$
The above simplifies to this
(4 + 4) + (a + a) = 8 + 2a

Presumably this isn't what you meant, which means you need to place parentheses around each numerator or denominator that has more than one term or factor, NOT just around the entire rational expression.

I'm guessing that what you really meant was this:
$\frac{4a}{a + 4} + \frac{a + 2}{2a}$
Using LaTeX, no parentheses are needed, but if you write the expression without LaTeX, it should probably be like so:
4a/(a + 4) + (a + 2)/(2a)
or possibly this:
4a/(a + 4) + (a + 2)a/2

Please tell us the expression you're starting with.

Mark, yes.
You guessed correctly at what I meant. (4a)/(a+4) + (a+2)/(2a).
It's much easier to write this on paper than it is to do it on a keyboard. LaTeX is brand new to me and I had no idea what it meant until now.

 

[cptstubing]

What are you trying to do "at the end" exactly?

You noted that the quadratic in the numerator does not factor, so I'm not sure what sort of simplification you think is possible. You've got to know when to stop!

If the homework says to combine and simplify, then I am certain they will have something to simplify. In the question so far, all I've done is combine terms to come out with (9a^2+6a+8) as a numerator, and (a+4)(2a) as a denominator.
The entire chapter has seen some simplification done at some point, usually by cancelling factors.
If there is actually no simplification necessary, I'm shocked and actually a bit confused.

 

[PeroK]

If the homework says to combine and simplify, then I am certain they will have something to simplify. In the question so far, all I've done is combine terms to come out with (9a^2+6a+8) as a numerator, and (a+4)(2a) as a denominator.
The entire chapter has seen some simplification done at some point, usually by cancelling factors.
If there is actually no simplification necessary, I'm shocked and actually a bit confused.

I can't say why this question was included, but part of completely understanding factorisation and simplification is knowing when it isn't possible. Like knowing when you can't factorise a quadratic.

 

[Ray Vickson]

Mark, yes.
You guessed correctly at what I meant. (4a)/(a+4) + (a+2)/(2a).
It's much easier to write this on paper than it is to do it on a keyboard. LaTeX is brand new to me and I had no idea what it meant until now.

You don't need to use LaTeX; writing 4a/(a+4) + (a+2)/(2a) is perfectly OK. It clear and unambiguous, and it means exactly what you wrote.

Note that you can write 4a/(a+4) instead of (4a)/(a+4), but you do need the parentheses around the second denominator; that is, you need to write (a+2)/(2a) rather than (a+2)/2a, because the latter is a little bit ambiguous---it could mean either $\frac{a+2}{2a}$ or $\frac{a+2}{2} a$.

 

[cptstubing]

Thank you all.