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How did they get these factors?!?

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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by smills518 » Sun Jan 27, 2013 3:28 pm
So I've been out of school for a LONG time and I'm finding I have to brush up on even the most basic algebra. In one of the books I'm using (an old Barron's book) they are discussing factoring algebraic equations and they say this:

(2x - 4) and (x + 2x^2 + x^3) are factors of (- 4x - 6x^2 + 2x^4)

when I tried to figure out how they got these factors, I tried to work backwards but got the following:

2x(- 2 - 3x + x^3) - and I could not factor the three term expression any further.

Could someone tell me how they managed to get the first set of factors because I can't figure it out?

Thanks in advance for tackling what seems like it will be a simple answer!
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by Tommy Wallach » Mon Jan 28, 2013 2:17 am
Hey Smills,

I think you're misinterpreting what's going on. They're just multiplying through.

(2x - 4) and (x + 2x^2 + x^3) are factors of (- 4x - 6x^2 + 2x^4)

2x * x = 2x^2
2x * 2x^2 = 4x^3
2x * x^3 = 2x^4
-4 * x = -4x
-4 * 2x^2= -8x^2
-4 * x^3 = -4x^3

That's -4x
2x^2 - 8x^2 = -6x^2
4x^3 - 4x^3 = 0
and 2x^4

so -4x -6x^2 + 2x^4

Hope that helps!

-t
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by Brent@GMATPrepNow » Mon Jan 28, 2013 6:58 am
smills518 wrote:So I've been out of school for a LONG time and I'm finding I have to brush up on even the most basic algebra. In one of the books I'm using (an old Barron's book) they are discussing factoring algebraic equations and they say this:

(2x - 4) and (x + 2x^2 + x^3) are factors of (- 4x - 6x^2 + 2x^4)

when I tried to figure out how they got these factors, I tried to work backwards but got the following:

2x(- 2 - 3x + x^3) - and I could not factor the three term expression any further.

Could someone tell me how they managed to get the first set of factors because I can't figure it out?

Thanks in advance for tackling what seems like it will be a simple answer!
For the GMAT, you do not need to know how to take an expression like -4x - 6x^2 + 2x^4 and factor it to get (2x - 4)(x + 2x^2 + x^3)

Yes, you do need to know how to factor out the Greatest Common factor to get: 2x(-2 - 3x + x^3), but that's it. Factoring -2 - 3x + x^3 would be out of scope for the GMAT.

You also need to know how to "expand." That is, multiply (2x - 4) by (x + 2x^2 + x^3) to get 2x^4 - 6x^2 - 4x


Factoring - 4x - 6x^2 + 2x^4 to get (2x - 4)(x + 2x^2 + x^3) requires something called the Remainder Theorem as well as a technique for dividing polynomials, two skills that the GMAT does not test.

I think the Barron's book is mainly demonstrating how to expand a product. That is, multiply (2x - 4) by (x + 2x^2 + x^3) to get 2x^4 - 6x^2 - 4x

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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