Polynomial long division

[kujac]

Hi, I'm trying to solve this problem.

(x²-3x-3)÷(x⁴-3x²+x+1)

I have this polynomial division where the degree of the dividend is smaller than the degree of the divisor.
How can I solve it in this case? (I have no problem solving it when its reversed; (x⁴-3x²+x+1)÷(x²-3x-3) )

 

[Cubist]

OP, are you sure that you didn't misread the index of the x² in the numerator (could it be x⁵ for example?)

There's a possibility that you were given a trick long division question. Maybe they are expecting you to write "this can't be done", or perhaps you could present it like this...

                                0
                  ---------------
x⁴ - 3x² + x + 1  )   x² - 3x  -3
                      0  +  0 + 0
                     ------------
                      x² - 3x  -3  (remainder)

Therefore the answer is:-
0 + (x² - 3x -3) / (x⁴ - 3x² + x + 1)
= (x² - 3x -3) / (x⁴ - 3x² + x + 1)
= original question

Was this a question that you were given, or have you come across this expression as part of some bigger question (and you thought that long division might be the way forward) ?

 

[skeeter]

[MATH]\dfrac{x^2-3x-3}{(x-1)(x^3+x^2-2x-1)}[/MATH]
is this part of a problem you haven't mentioned?

 

[kujac]

[MATH]\dfrac{x^2-3x-3}{(x-1)(x^2-2x-1)}[/MATH]
that's about it ... why are you trying to divide in the first place? Is this part of a problem you haven't mentioned?

OP, are you sure that you didn't misread the index of the x² in the numerator (could it be x⁵ for example?)

There's a possibility that you were given a trick long division question. Maybe they are expecting you to write "this can't be done", or perhaps you could present it like this...

                                0
                  ---------------
x⁴ - 3x² + x + 1  )   x² - 3x  -3
                      0  +  0 + 0
                     ------------
                      x² - 3x  -3  (remainder)

Therefore the answer is:-
0 + (x² - 3x -3) / (x⁴ - 3x² + x + 1)
= (x² - 3x -3) / (x⁴ - 3x² + x + 1)
= original question

Was this a question that you were given, or have you come across this expression as part of some bigger question (and you thought that long division might be the way forward) ?

no, the index is right (x²). It was part of bigger task (more polynomial divisions, multiplications etc.). Perhaps it's tricky question but I'm just wondering if it's possible to solve because I have never come across to problem like this.

 

[Steven G]

When the degree of what you are dividing is smaller than the degree of what you are dividing well that is what a remainder is called.

If you are correct that you never came across this then you never did polynomial division. The reason is that with polynomial division there is an end to the process and the end is exactly when the degree of the dividend is smaller than the degree of the divisor