What are the Basics of Factoring and Understanding Proofs?

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In summary, the conversation discusses using arbitrary variables to prove that if a number F divides A, then it also divides mA. They also mention a theorem stating that the HCF of the numerator and denominator of a fraction must be a factor of their sum. The example given is used to demonstrate this theorem and raise questions about how to determine the HCF without dividing the polynomials.
  • #1
Miike012
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Homework Statement



The proof has to do with factoring... What do all the letters mean?
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Homework Equations





The Attempt at a Solution

 

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  • #2
F is an arbitrary factor.
A is an arbitrary number.
B is an arbitrary number, distinct from A and F.
a and b are arbitrary numbers, used to make it easier to understand in which case for they are being used.
n and m are arbitrary numbers.

Take the first case: Given F divides A (that is, if you divide a number A by F, the result is a natural number), then prove that it divides mA (that is, prove that F divides any multiple of A).
They then use a general case that A=aF...A is a multiple of F (since F divides A, it must be a factor and thus A a multiple) to show from there.

Does that make the second case clearer?
 
  • #3
Miike012 said:

Homework Statement



The proof has to do with factoring... What do all the letters mean?
I added an attachment.

Homework Equations





The Attempt at a Solution


The variables can be anything: ie they can have any value. The only restriction is that the variables 'a' and 'b' have to be non zero whole numbers. Besides that the variables can be any whole number you like.
 
  • #4
Here is something else that I just came across...
the HCF of the numerator and denominator must be a factor of their sum (Thats odd isn't it? Is there some proof to this?)

The example:( 3x^3 - 13x^2 +23x -21) / 15x^3 - 38x^2 -2x +21)
Sum of num and den. = 18x^3 - 51x^2 +21x = 3x(3x-7)(2x-1)

This is where I get confused... The book says " if there is a common divisor is is clearly 3x - 7." My problem is, the book makes it sound like it is the obvious choice... How would you know that 3x-7 is the HCF without dividing 3x-7 then 3x then 2x-1 sepperatly into the denominator and numerator?
 
  • #5
Shot in the dark:

Maybe because the coefficient of the highest degree term in each polynomial is not divisible by 2 (precluding 2x-1) and because each polynomial has a term of degree 0 so so much for 3x?
 
  • #6
Does this "theorem" have a name? I've never hurd of this before? Its interesting.
 

FAQ: What are the Basics of Factoring and Understanding Proofs?

What does it mean to "not understand proof"?

Not understanding proof refers to the difficulty in comprehending the logical steps and reasoning behind a mathematical or scientific proof. It can also involve struggling to grasp the significance or implications of a particular proof.

Why is understanding proof important in science?

In science, understanding proof is crucial because it allows us to validate and verify theories and hypotheses. A proof provides evidence that supports or refutes a claim, and without understanding it, we cannot fully grasp the validity of a scientific concept or idea.

What are some common reasons for not understanding proof?

There are several possible reasons for not understanding proof, such as a lack of prior knowledge or understanding of the underlying concepts, difficulty with abstract thinking, and inadequate explanation or presentation of the proof.

How can one improve their understanding of proof?

Improving understanding of proof requires a combination of practice, patience, and seeking additional resources such as textbooks, online tutorials, or consulting with a mentor or teacher. It can also be helpful to break down the proof into smaller, more manageable steps and to actively engage with the material by asking questions and attempting to explain the reasoning to someone else.

Is it normal to struggle with understanding proof?

Yes, it is entirely normal to struggle with understanding proof, especially for complex or advanced concepts. It is essential to remember that understanding takes time and effort, and seeking help and resources is always a valuable step in improving understanding.

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