What is the process for finding the LCD of rational expressions and equations?

[awholenumber]

OK , I have three questions
I was a computer science student , so i am a bit confused about some terms .

What is a variable x ?

x is a quantity
x is binary
x is a point mass object
x is digital logic
x is electrons The other is when you try to find the "pairs of factors" of a number . You can either find its composite factor or prime factors . right ?

I hope that is what it is really called , "pairs of composite factors" and "pairs of prime factors"

Third one is when you try to solve a rational equation involving polynomials , you try to factor the polynomials in the numerator and the denominator , then find the LCM from the factors , right ?

Can somebody give me an example of that ?

 

[mfb]

What is a variable x ?

It depends on the context. x is just a letter that can be used for everything.

The other is when you try to find the "pairs of factors" of a number . You can either find its composite factor or prime factors . right ?

The pairs are sets of two numbers that, when multiplied, give the number. It doesn't matter if these numbers are prime or not. If you find pairs for 8, for example, then you have (1,8) and (2,4). Just one of the four numbers in the pairs is a prime number.

Third one is when you try to solve a rational equation involving polynomials , you try to factor the polynomials in the numerator and the denominator , then find the LCM from the factors , right ?

Why should you? If you want to add multiple fractions, finding the least common multiple of the denominators can be useful.

 

[awholenumber]

Thanks for the reply ,

In the first equation , why is 4x² the LCM ? What are the steps involved to find LCM like that ?

Same question here , i don't understand taking the LCM part properly ?

 

[awholenumber]

https://2012books.lardbucket.org/books/beginning-algebra/s10-rational-expressions-and-equat.html

It is not always the case that the LCD is the product of the given denominators. Typically, the denominators are not relatively prime; thus determining the LCD requires some thought. Begin by factoring all denominators.
The LCD is the product of all factors , with the highest power

 

[QuantumQuest]

For two numbers, their LCM is the smaller positive integer divisible by both. If you also have variables (like $x$) at various positive powers then you have to take the one with the biggest exponent. These concepts expand for more numbers and / or powers of variables.

For instance, in this example you ask about

In the first equation , why is 4x2 the LCM ? What are the steps involved to find LCM like that ?

$4$ is the smaller positive integer divisible by all $4, 1$ and $2$. Now, $x^2$ is the biggest power of $x$, so combining, $4x^2$ is the LCM.

For more see Wikipedia LCM.

 

[awholenumber]

Thanks , Steps to Find the LCM of Two or More Rational Expressions

1. Factor all denominators completely.
2. The LCM is the product of unique prime factors from the denominators,
where each factor is raised to the highest power to which it appears in
any denominator.

Can somebody show me the steps involved ?

 

[Merlin3189]

In the first equation , why is 4x2 the LCM ? What are the steps involved to find LCM like that ?

For numbers, the LCM is the smallest number that they are all a factor of.
The way to find it is to list all the factors of each number and then make the smallest number that has all those factors. The best way to do this is to break them into their prime factors. (I'll use dot for multiply, so that I can use x for exe.)

Find LCM of 8, 12, 18, 30 Factors of 8=[2.2.2] of 12=[2.2.3] of 18=[ 2.3.3] and of 30=[2.3.5]
So we need a number which has prime factors of 2.2.2 . 3.3 . 5 because all the constituent numbers can be made from subsets of these factors.
So LCM = 8.9.5=360

For algebraic expressions the idea is the same: find the prime factors of all the parts and make a term that contains all of them
4x = 2.2.x , x² = x.x , 2x² =2.x.x
So LCM = 2.2.x.x = 4x²
That was a bit easy, so try LCM of 6a²b , 9abc² and 24ab³
6a²b = 2.3.a.a.b , 9abc² =3.3.a.b.c.c , 24ab³ =2.2.2.3.a.b.b.b
So LCM = 2.2.2.3.3.a.a.b.b.b.c.c = 72a²b³c²

As for why do you find the LCM, that's just what you need to sort out fractions. Multiplying a fraction by its denominator give a whole number -eg. $\frac 3 4 \times 4 = 3$ or $\frac 3 4 \times 20 = 3 \times 5 = 15$

Say you had $\frac a 6 + \frac {5} {8} = \frac { 11} {12 }$ so find the LCM of 6, 8 and 12 which is 24 and multiply everything by 24 to get rid of all the fractions
So. $24 \frac a 6 +24 \frac {5} {8} =24 \frac {11} {12 }$
So. $4a + 15 = 22$
So. $4a = 7$
So. $a = \frac{7}{4}$

In your example, the fractions have numerators, (x +3) , (x - 3) , (x² - 9)
So their prime factors are (x +3) = (x + 3) , (x - 3) = (x - 3) , and (x² - 9) = (x + 3).(x - 9)
So LCM = (x + 3).(x - 3) and you multiply each fraction by this,as shown in the eg. which gets rid of all the fractions.

Edit: sorry, 2 more posts appeared while I was wrestling with the Latex!

 

[awholenumber]

Thanks a lot for that really long explanation .

I understand this part properly
1. Factor all denominators completely

4x = 2.2.x , x² = x.x , 2x² =2.x.x

We have seen in Class VII that in algebraic expressions, terms are formed as products of
factors. For example, in the algebraic expression 5xy + 3x the term 5xy has been formed
by the factors 5, x and y, i.e.,
5xy = 5 * x * y
Observe that the factors 5, x and y of 5xy cannot further
be expressed as a product of factors. We may say that 5,
x and y are ‘prime’ factors of 5xy. In algebraic expressions,
we use the word ‘irreducible’ in place of ‘prime’. We say that
5 × x × y is the irreducible form of 5xy. Note 5 × (xy) is not
an irreducible form of 5xy, since the factor xy can be further
expressed as a product of x and y, i.e., xy = x × y.

So ,

2. The LCM is the "product" of unique ( being the only one of its kind; unlike anything else )"pairs of irreducible factors" from the denominators,
where each factor is raised to the highest power to which it appears in
any denominator.

4x = 2.2.x , x² = x.x , 2x² =2.x.x
So LCM = 2.2.x.x = 4x²

 

[Mark44]

No one else has responded in depth to this part of your first post, so I will take a shot at it.

What is a variable x ?

x is a quantity
x is binary
x is a point mass object
x is digital logic
x is electrons

In terms of mathematics, a variable is some number that is an unknown quantity, that can take on different values. In the context of algebra, a typical problem is to find the value of some variable so that an equation or inequality represents a true statement.

"Binary" can have at least two meanings:
1) Having one of two possible values, such as "true" or "false", or "up" or "down," "on" or "off," 1 or 0, or "black" or "white," -- whatever, where there is a choice of two states.
2) A representation of a number in base-2. The base-10 (or decimal) number 18 has 10010 as its base-2 representation. Computers generally deal with numbers in their binary representation.

"x is a point mass object" - All this does is associate a name ("x") with some object, all of whose mass is considered to be at a single point. For the sake of simplicity, engineers and physicists sometimes assume that the masses involved in structures are point masses.

"x is digital logic" - This is pretty much meaningless.

"x is electrons" - Also meaningless, unless x happens to represent some attribute of the electrons, such as how many there are, their total charge, their total mass, or some other unstated trait of electrons.

 

[jbriggs444]

In terms of mathematics, a variable is some number that is an unknown quantity, that can take on different values. In the context of algebra, a typical problem is to find the value of some variable so that an equation or inequality represents a true statement.

One pithy statement is that "In computer science, we assign values to variable names. In mathematics, we assign variable names to values".

In BASIC, one could say: "Let x = 8" . That would be an assignment statement. We store the numeric value 8 in the memory cell labelled x.

In Mathematics, one can say: "Let x be a real number". This is like closing our eyes, taking a number, wrapping it in an envelope and writing "x" on the outside of the envelope. We may not know what number is in the envelope, but we have a label for it. We can use that label in expressions and we can manipulate it using algebra.

[Unlike @Mark44, I tend to regard Mathematics as not allowing the contents of the metaphorical envelope to be altered. Instead, we can rip up the old envelope, create a new one for the next number and scribble the old variable name on the new envelope. No biggie -- it's just a metaphor]

There are different contexts in which a variable name may be encountered.

Algebra:

In algebra, one may write down an equation such as $x^2 + 2x + 1 = 0$. This amounts to an assertion about the unknown value named x. As @Mark44 suggests, one usually wants to "solve" this equation, finding the possible values of x for which the equation is true.

Function definitions:

One can write down something like: "let f(x) = x² + y". This is a stylized notation used to define a function. The left hand side is the the function name and its dummy parameter list. The right side is an expression that uses those dummy parameters. In this case we are defining a function with one parameter. The y here is not an argument to the function. It is considered to be a constant. As a result, we cannot know exactly what one-parameter function has been defined unless we can come up with a value for y.

Quantifiers:

One can provide context for a variable name by saying things like "for all real numbers x, x² > 0". The "for all" is called a quantifier -- the "universal" quantifier. The symbol "∀" means "for all" and is a shorthand way of using the universal quantifier. The parallel in computing would be something like:

result = true
for all x in the real numbers
  if x^2 <= 0 then result = false
end for
print result

Similarly, one can say "there is a real number x such that x² = 0". This "there exists" is called a quantifier -- the "existential" quantifier. The symbol "∃" means "there exists" and is a shorthand way of using the existential quantifier. The parallel in computing would be something like:

result = false
for all x in the real numbers
  if x^2 = 0 then result = true
end for
print result

 

[awholenumber]

Thanks for the replies .
Mentor note: Unrelated images have been deleted.
So ,
x is electrons
x is a quantity
x is binary
x is a point mass object
x is digital logic

Is this an OK definition ?

Steps to Find the LCM of Two or More Rational Expressions

1. Factor all denominators completely.
2. The LCM is the product of unique prime factors from the denominators,
where each factor is raised to the highest power to which it appears in
any denominator.

The LCM is the "product" of unique ( being the only one of its kind; unlike anything else )"pairs of irreducible factors" from the denominators,
where each factor is raised to the highest power to which it appears in
any denominator.

 

[Mark44]

Thanks for the replies .
Mentor note: Unrelated images have been deleted.
So ,
x is electrons
x is a quantity
x is binary
x is a point mass object
x is digital logic

Is this an OK definition ?

A definition for what?
Did you read what I wrote earlier?

Steps to Find the LCM of Two or More Rational Expressions

1. Factor all denominators completely.
2. The LCM is the product of unique prime factors from the denominators,
where each factor is raised to the highest power to which it appears in
any denominator.

 

[Mark44]

In Mathematics, one can say: "Let x be a real number". This is like closing our eyes, taking a number, wrapping it in an envelope and writing "x" on the outside of the envelope. We may not know what number is in the envelope, but we have a label for it. We can use that label in expressions and we can manipulate it using algebra.

[Unlike @Mark44, I tend to regard Mathematics as not allowing the contents of the metaphorical envelope to be altered.

I don't think we really disagree. Starting with "Let x be a real number..." - the value of x is completely undefined, other than being some real number.
If we add a qualification, "...such that $x^2 + 2x + 1 = 0$, then we're talking about exactly one real number, whose value is unknown until we solve the quadratic equation.

 

[awholenumber]

A definition for what?

To calculate an LCM for a rational function, follow these steps:
1. Factor all denominator polynomials completely.
2. Make a list that contains one copy of each factor, all multiplied together.
3. The power of each factor in that list should be the highest power that factor is
raised to in any denominator.
4. The list of factors and powers you generated is the LCM.

 

[Mark44]

To calculate an LCM for a rational function, follow these steps:
1. Factor all denominator polynomials completely.
2. Make a list that contains one copy of each factor, all multiplied together.
3. The power of each factor in that list should be the highest power that factor is
raised to in any denominator.
4. The list of factors and powers you generated is the LCM.

Yes, this looks fine, but you wrote this:

So ,
x is electrons
x is a quantity
x is binary
x is a point mass object
x is digital logic

Is this an OK definition ?

I assumed that by "OK definition" you were asking about what was immediatedly above it.

 

[awholenumber]

OK , Thanks

 

[Mathivanan]

OK , I have three questions
I was a computer science student , so i am a bit confused about some terms .

What is a variable x ?

x is a quantity
x is binary
x is a point mass object
x is digital logic
x is electronsThe other is when you try to find the "pairs of factors" of a number . You can either find its composite factor or prime factors . right ?

I hope that is what it is really called , "pairs of composite factors" and "pairs of prime factors"

Third one is when you try to solve a rational equation involving polynomials , you try to factor the polynomials in the numerator and the denominator , then find the LCM from the factors , right ?

Can somebody give me an example of that ?

The first question:what is a variable 'x'?
I like to explain this question with an example. Suppose you deposit with me 'some' money I will give you two times of that after 3 years. Now how do you answer this question: how much money will I give you after 3 years? It depends upon the money you deposit with me. Right. That is, if you deposit with me 'x' amount of money then after two years I will give you '2*x' amount of money. 'x' can take any value. 'x' is a variable quantity.

 

[Mathivanan]

I don't think we really disagree. Starting with "Let x be a real number..." - the value of x is completely undefined, other than being some real number.
If we add a qualification, "...such that $x^2 + 2x + 1 = 0$, then we're talking about exactly one real number, whose value is unknown until we solve the quadratic equation.

There is a difference. The value of the term x^2+2*x+1 is a variable, because x is a variable. However, x^2+2*x+1=0 is an equation, which fixes the value for 'x'. That is, for what value/s of 'x' the term equals zero.

 

[Mark44]

There is a difference. The value of the term x^2+2*x+1 is a variable, because x is a variable. However, x^2+2*x+1=0 is an equation, which fixes the value for 'x'. That is, for what value/s of 'x' the term equals zero.

You're not saying anything different from what I said.

 

[Mathivanan]

You're not saying anything different from what I said.

I haven't tried to say something different from what you said. The full conversation was not shown to me. It is messy. On the other hand, the 'difference' means the difference between the term and the equation.

 

[Mark44]

I haven't tried to say something different from what you said. The full conversation was not shown to me. It is messy. On the other hand, the 'difference' means the difference between the term and the equation.

Indeed it is very messy.

The value of the term x^2+2*x+1

A minor correction -- This expression has three terms: x², 2x, and 1. Terms are expressions that are added or subtracted.

 

[awholenumber]

Mathivanan ,

Thanks for the reply .
I was mostly after finding LCD of the denominators of rational expressions and equations . I have more clarity now .

STEP 1: Factor each denominator into the product of its lowest terms, and express any repeating
numbers as powers.

STEP 2: List all the different numbers that appear for each denominator. If one number is common for
any of the denominators, only list that number once to the highest power that it appears.

STEP 3: The product of the factors listed in Step 2 will be the LCD.