How Do I Factor This Expression?

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Discussion Overview

The discussion revolves around the factoring of the expression 3(x + h)^4 - 48(x + h)^2. Participants explore the steps involved in factoring, definitions of factoring, and its relationship to multiplication and division.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant proposes factoring out 3(x + h)^2 from the expression, leading to the form 3(x + h)^2[(x + h)^2 - 16].
  • Another participant agrees with the initial factoring and confirms the steps taken.
  • A participant questions whether factoring can be defined as the opposite of multiplication, leading to a discussion about the relationship between factoring and division.
  • One participant suggests that division is the inverse of multiplication, while noting that factoring is related to division through the example of the difference of squares.
  • Several participants seek a non-technical definition of factoring, with one suggesting it involves finding what to multiply together to obtain an expression.

Areas of Agreement / Disagreement

There is agreement on the initial factoring steps proposed by the first participant. However, there is no consensus on the definitions and relationships between factoring, multiplication, and division, as participants express differing views.

Contextual Notes

The discussion includes varying definitions and interpretations of factoring, which may depend on participants' mathematical backgrounds and understanding. Some definitions may lack precision or clarity.

Who May Find This Useful

Readers interested in algebra, particularly in understanding factoring techniques and definitions, may find this discussion beneficial.

mathdad
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Factor the expression.

3(x + h)^4 - 48(x + h)^2

I believe I can factor out 3(x + h)^2.

If so, we then have the following:

3(x + h)^2[(x + h)^2 - 16]

I think (x + h)^2 - 16 is the difference of two perfect squares.

If so, then it factors out to be [(x + h) - 4][(x + h) + 4].

Answer: 3(x + h)^2[(x + h) - 4][(x + h) + 4].

Correct?
 
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Perfect. (Yes)
 
Great! Factoring is fun. Can I safely define factoring as the opposite of multiplication?
 
I would say division is the inverse of multiplication, but factoring is certainly related to division. For example, we know:

$$a^2-b^2=(a+b)(a-b)\,\therefore\,a-b=\frac{a^2-b^2}{a+b}$$
 
Without using math jargon, what is the simplest definition of factoring?
 
RTCNTC said:
Without using math jargon, what is the simplest definition of factoring?

Finding what to multiply together to get an expression. It is like "splitting" an expression into a multiplication of simpler expressions.
 
MarkFL said:
Finding what to multiply together to get an expression. It is like "splitting" an expression into a multiplication of simpler expressions.

Thank you.
 

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