Division of polynomials

burgerandcheese

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Question: Find the factors of p(x) = 3x3 + 4x2 + 5x - 6

Here's what it says in the book I'm reading
11854

I really don't understand the part where it says sx - t is not really a different factor from -sx + t. So you need consider only positive values of s. Can someone please explain it to me?

And as for the one below that, how did they conclude that 3 would be a common factor of the coefficients of p(x)? Is it because 3x ± 3 = 3(x ± 1) and 3x ± 6 = 3(x ± 2)?
 
Question: Find the factors of p(x) = 3x3 + 4x2 + 5x - 6

Here's what it says in the book I'm reading
View attachment 11854

I really don't understand the part where it says sx - t is not really a different factor from -sx + t. So you need consider only positive values of s. Can someone please explain it to me?

And as for the one below that, how did they conclude that 3 would be a common factor of the coefficients of p(x)? Is it because 3x ± 3 = 3(x ± 1) and 3x ± 6 = 3(x ± 2)?
sx - t is not really a different factor from -sx + t. So you need consider only positive values of s.

Do you see that:

s*x - t = (-1) * (-s*x + t)

how did they conclude that 3 would be a common factor

Yes ... your logic is correct.
 
Yes, sx-t is different from -sx+t. The text above says that sx-t and -s+t are not really different meaning that if sx-t is a factor then -s+t is also a factor.

Here is an example: (x-5)(x+3) = (5-x)(-x-3)
 
Yes, sx-t is different from -sx+t. The text above says that sx-t and -s+t are not really different meaning that if sx-t is a factor then -s+t is also a factor.

Here is an example: (x-5)(x+3) = (5-x)(-x-3)

I get your example, but I still don't understand :(
 
I am agreeing with you that sx-t and -sx+t are different.
Suppose you want to factor x^2+5x+6
You can x^2 by multiplying x and x OR -x and -x. That is x^2 +5x+6 = (x+2)(x+3) OR x^2+5x+6 = (-x-2)(-x-3).
Now your book is simply saying that you can consider getting x^2 by multiplying x and x (and getting (x+2)(x+3)) OR by multiplying -x and -x (and getting (-x-2)(-x-3)) BUT there is NO NEED to consider both.
A better example will be -6x^2+13x-6. You can get -6x^2 by multiplying -3x and 2x (and getting (-3x+2)(2x-3)) OR by multiplying 3x and -2x ( and getting (3x-2)(-2x+3) but you do NOT need to consider both -3x and 2x AND 3x and -2x.
Do you understand now? This will allow you to look at only half of the possible factorings.
 
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