How Do You Simplify a Cubic Polynomial?

[Patdon10]

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.

 

[Mentallic]

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?

 

[HallsofIvy]

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.

 

[Mentallic]

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

 

[SammyS]

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