How Do You Simplify a Cubic Polynomial?

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Homework Help Overview

The discussion revolves around the simplification and factorization of the cubic polynomial -x^3 - 3x^2 - 4x - 2. Participants are exploring the correct approach to factor this polynomial and questioning the validity of a proposed factorization.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the need to factor the polynomial rather than simplify it. There is a focus on identifying rational roots and verifying proposed factors. Some participants question the correctness of the factorization given in the original post.

Discussion Status

The discussion is active, with participants providing insights into checking for roots and the implications of incorrect factorization. There is a recognition of the need to divide the polynomial by a valid factor, and some guidance has been offered regarding the process of finding roots.

Contextual Notes

Participants note that the original factorization provided may be incorrect, as it does not satisfy the polynomial equation when evaluated at certain points. There is an emphasis on the importance of checking roots before proceeding with factorization.

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Homework Statement



simplify -x^3 - 3x^2 - 4x - 2
It is equal to -(x-1)(x^2 + 2x + 2)

Not sure how to get that answer, nor how to start it.
 
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Patdon10 said:

Homework Statement



simplify -x^3 - 3x^2 - 4x - 2
It is equal to -(x-1)(x^2 + 2x + 2)

Not sure how to get that answer, nor how to start it.

The word you're looking for is to factorize it, not simplify it. You can't simplify it in the sense that you can simply \frac{\sin(x)}{\cos(x)} to become \tan(x) for example.

Ok so first of all, in order to factorize that cubic you'll need to know one of its roots. Do you know how to check if a polynomial has rational roots?
 
Patdon10 said:

Homework Statement



simplify -x^3 - 3x^2 - 4x - 2
It is equal to -(x-1)(x^2 + 2x + 2)

Not sure how to get that answer, nor how to start it.
Good! Because you shouldn't get that "answer". It is wrong.

Setting x= 1 in that polynomial gives -(1)- 3(1)- 4(1)- 2= -(1+3+ 4+ 2)= -10, not 0. Since x= 1 does NOT make that polynomial 0, x- 1 is NOT a factor. -x^3- 3x^2- 4x- 2 is NOT equal to -(x- 1)(x^2+ 2x+ 2).

However, setting x= -1 gives -(-1)- 3(1)- 4(-1)- 2= 1- 3+ 4- 2= 0 so x-(-1)= x+ 1 is a factor. Divide -x^3- 3x^2- 4x- 2 by x+1 to get the other factor.
 
HallsofIvy said:
Good! Because you shouldn't get that "answer". It is wrong.

Setting x= 1 in that polynomial gives -(1)- 3(1)- 4(1)- 2= -(1+3+ 4+ 2)= -10, not 0. Since x= 1 does NOT make that polynomial 0, x- 1 is NOT a factor. -x^3- 3x^2- 4x- 2 is NOT equal to -(x- 1)(x^2+ 2x+ 2).

However, setting x= -1 gives -(-1)- 3(1)- 4(-1)- 2= 1- 3+ 4- 2= 0 so x-(-1)= x+ 1 is a factor. Divide -x^3- 3x^2- 4x- 2 by x+1 to get the other factor.

Ah yes I shouldn't noticed that at quick glance since -x^3 - 3x^2 - 4x - 2 = -(x^3+3x^2+4x+2) which should have all same sign coefficients in its factors
 
Patdon10 said:

Homework Statement



simplify -x^3 - 3x^2 - 4x - 2
It is equal to -(x-1)(x^2 + 2x + 2)

Not sure how to get that answer, nor how to start it.
-(x-1)(x^2 + 2x + 2)

is equivalent to:

-x^3 -x^2 + 2
 

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