remove a common factor

hhtt21

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"In some polynomials pairs of terms have a common factor that can be removed, as in the following examples. This process is called factoring by grouping,
and uses the distributive laws repeatedly."

Examples: y^2+3y+4y+12=y(y+3)+4(y+3)

What does "remove" mean above? I understand "remove" by very different things than above.

Thank you.
 
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GoesStation

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If the first expression is y^2+3y+4Y+12, the common factor y+3 has been removed. I don't know whether this is standard terminology in algebra, but it makes sense to me.

The expression on the right can be further simplified to (y+4)(y+3).
 

hhtt21

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However, this book uses 'remove' to mean: take common factors outside of the brackets, e.g., (y + 3)[y + 4]. After you learn the topic just say: "Factor the following expression" and you won't sound amateurish.

I understand remove means pick common factors out. Is this a common meaning for remove? Do you mean remove and factoring do not collocate well? What word would you offer instead of remove for this context?

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hhtt21

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Look! Forget 'remove'. You're not removing anything. You might be moving something but you are removing nothing. They do not collocate well, to use your expression. Factor this for me; I'll wait: X^2 + 4X + 4.

X^2+4X+4=(X+2)(X+2)=(X+2)^2

I am wondering what will happen related to "remove"

Thank you.
 

YAMATO2201

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I don't know whether this is standard terminology in algebra, but it makes sense to me.
I would use "factor out" instead of "remove".
 

hhtt21

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I would use "factor out" instead of "remove".

I think factor out is the name of the whole process, not part of it but remove refers to the something part of factor out.

Thank you.
 

YAMATO2201

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I think factor out is the name of the whole process, not part of it but remove refers to the something part of factor out.
"The first step in completely factoring a polynomial is to remove (factor out) any common factors, as shown in the next example."
(Larson, R., College Algebra: Real Mathematics, Real People, 7th ed., p.26)

"To factor the expression 6x^4-6x, for example, you first factors out the common factor, 6x, and then ..."
(Sterling M. J., Algebra II For Dummies, 2nd ed., p.16)

"Then we factor out the greatest common divisor of the coefficients of ..."
(Artin, M., Algebra, 1st ed., p.399) [Artin's Algebra was the required text for Harvard's Math 123.]

"All we have done is to factor out a "common denominator" for the coefficients of f."
(Hungerford, T. W., Algebra, p.314)

In each of these quotations above, factor out does not mean factor a polynomial into irreducible factors.
 
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