Rational functions: combine and simplify terms

Click For Summary

Homework Help Overview

The discussion revolves around the simplification of a rational function involving the expression (4a/(a+4)) + ((a+2)/(2a)). Participants are exploring how to combine and simplify the terms while addressing the correct placement of parentheses to avoid ambiguity in the expression.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the process of combining the fractions and express uncertainty about the simplification steps, particularly regarding the factoring of the resulting quadratic numerator. Questions arise about the correct interpretation of the expression due to parentheses placement.

Discussion Status

There is an ongoing exploration of the expression's structure and the simplification process. Some participants have provided clarifications on the proper use of parentheses, while others express confusion about the potential for further simplification of the quadratic term.

Contextual Notes

Participants note the need to state restrictions on the variable and express concern about the possibility of canceling factors. There is a shared understanding that the homework requires both combining and simplifying the expression, but clarity on whether further simplification is possible remains a point of contention.

cptstubing
Messages
37
Reaction score
3

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.
 
Physics news on Phys.org
cptstubing said:

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?
 
PeroK said:
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.
 
cptstubing said:

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.
cptstubing said:

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.
 
cptstubing said:
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!
 
  • Like
Likes   Reactions: cptstubing
cptstubing said:
=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.
 
  • Like
Likes   Reactions: cptstubing
Mark44 said:
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.
 
PeroK said:
What are you trying to do "at the end" exactly?
PeroK said:
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.
 
cptstubing said:
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.
 
  • Like
Likes   Reactions: cptstubing
  • #10
cptstubing said:
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##.
 
  • Like
Likes   Reactions: cptstubing
  • #11
Thank you all.
 

Similar threads

Prove that [a/b]+[2a/b]+....+[(b-1)a/b]=(a-1)(b-1)/2
  • · Replies 5 ·
Replies
5
Views
4K
Graphical Transformations and Finding the Equation of a Curve
  • · Replies 3 ·
Replies
3
Views
2K
How to Simplify a Rational Function
  • · Replies 2 ·
Replies
2
Views
2K
Algebra II Simplifying Rational Algebraic Expressions
  • · Replies 13 ·
Replies
13
Views
4K
Simplifying rational functions with common factors
  • · Replies 4 ·
Replies
4
Views
2K
Deriving the quadratic formula.
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Determine PDE Boundary Condition via Analytical solution
  • · Replies 1 ·
Replies
1
Views
3K
Finding a polynomial function given zeros
  • · Replies 3 ·
Replies
3
Views
9K
Antiderivative of a rational function
  • · Replies 4 ·
Replies
4
Views
2K
How to Simplify Fractions with Common Factors
  • · Replies 8 ·
Replies
8
Views
3K