How do I correctly solve this polynomial long division problem?

[Kys91]

I am trying to solve:

$$\frac{x^6+6x^3-2x^5-7x^2-4x+6}{x^4-3x^2+2}$$

Using the polynomial long division algorithm.

I order first the terms of the divident, and leave one blank space between -2x^5 and +6x^3

My problem is, I first put x^2 to the quotient, so I get x^4 * x^2 = x^6, but then I multiply x^2 * -3x^2 = -3x^4, which can't be subtracted with -2x^5.

I have tried playing around but with no success.

Thanks

 

[Bohrok]

You also need to leave blank spaces in your divisor. Better yet, where there should be blank spaces, put the missing term with a coefficient of 0. In your divisor, put 0x³. Multiply that by x² and you get 0x⁵ which should be easy to subtract from -2x⁵. Try doing that with all missing terms in the dividend and divisor because it can be easy to miss something with all those terms.

 

[symbolipoint]

Your dividend takes the form as x^6 -2*x^5 +0*x^4 +6*x^3 -7*x^2 -4x +6
Restating your divisor as x^4 +0*x^3 -3*x^2 +0*x +2

Notice every degree of x must be shown. This is like keeping "place value" in "integers" were those to be divided, but now we are keeping track of powers of x, not powers of 10.

Your first partial division was (x^6)/(x^4)=x^2, this was good. Now, how much complete divisor do you subtract? Find this by performing multiplication:
(x^2)*(x^4 +0*x^3 -3*x^2 +0*x +2)=x^6 +0*x^5 -3*x^4 +0*x^3 +2*x^2

Now you write x^6 +0*x^5 -3*x^4 +0*x^3 +2*x^2 in proper alignment under the dividend and perform your first subtraction. Now, continue from here.

 

[Kys91]

Thank you, much easier with putting zeros.

I also noticed that I messed up a lot doing this: 0x^4 - (-3x^4) = +3x^4, I forgot to change the sign many times.

Your help is very much appreciated.