How do I divide this polynomial (using division algorithm)?

Question

I have to divide x + 2 into f(x) = x^3 + 9x^2 + 7x + 3 and then write my answer in the form given by the Division Algorithm; x^3 + 9x^2 + 7x + 3 = q(x)(x+2) + r(x) for some polynomials q(x) and r(x).
How do I do this?
Thanks very much!

Answer 1

Wikipedia is your friend:
http://en.wikipedia.org/wiki/Polynomial_long_division

Answer 2

Everything shifts to the left when I save this no matter what I do, so if you don't get it, just send me a message, and i'll send you something different. *Sorry*

You just divide x+2 into f(x) using the same, basic set-up you would use for regular long division:
________________
x + 2 ) x^3 + 9x^2 + 7x + 3

Divide x into the first monomial x^3, which using the your exponential rules [x^3/x = x^(3-1)] equals x^2. Place the solution above the monomial. Next, you need to multiply both the both the x and the 2 by your solution [x^2(x+2)] placing this new polynomial below the coinciding monomials and then subtract. See below [Sorry if it's confusing, It's hard to explain on clearly on the computer]:

_x^2______________
x + 2 ) x^3 + 9x^2 + 7x + 3
- x^3 - 2x^2
0 + 7x^2

Now, hopefully it's starting to look like regular old long division except the extra monomial. Next, you will once again divide the x from x+2, but this time you divide it into the difference you calculated and not the original equation [7x^2]. Once you multiply x+2 by your new solution, you will subtract the product from the original equation again. The only change is bringing down the 7x just like normal long division. See Below:

x^2 + 7x - 7
________________
x + 2 ) x^3 + 9x^2 + 7x + 3
- x^3 - 2x^2 | |
~~~~~~~~~ | |
0 + 7x^2 + 7x |
- 7x^2 - 14x |
~~~~~~~~~ |
0 - 7x + 3
+ 7x + 14
~~~~~~~~
0 + **17**

The polynomial doesn't go into f(x) evenly, so it has a remainder of 17. The proper equation for this problem is:

f(x) = q(x)g(x) + r(x), where q(x) = factor of f(x) & g(x);
r(x) = remainder [r/g(x)]

x^3 + 9x^2 +7x + 3 = (x^2 + 7x - 7)(x + 2) + 17/(x+2)
And that's the solution AND x= - 2 is not a factor of f(x) because of the remainder (if it was 0 then it would be a factor)!...

There is a much quicker and easier way to divide polynomials called synthetic division. I don't know if you are familiar with it, but I thought I'd tell you just in case. It's RIDICULOUSLY more simple than long division. You just have to remember that *g(x) = x - k* or in simpler words you always switch the sign of the number (k) by solving for x [In this case x+2 ---> x= - 2 or 'k':

Just place 'k' & all of f(x)'s coefficients in the order shown below going from the highest exponent to the lowest. Don't forget that if your missing a monomial in this order you must place a 0 in it's missing spot!! For instance, x^3 + 8 ---> You must place the missing 0 coefficients for each missing exponent; 1 0 0 8 (Replacing x^3 + 0x^2 + 0x + 8)...

___
- 2 | 1 9 7 3

________________
1

Bring the first coefficient down as shown, then multiply it by 'k' or - 2 and place the product under the next coefficient. After, sum these two numbers and repeat the same process for that solution and everyone thereafter:

___
- 2 | 1 9 7 3
- 2 -14 14
_________________
1 7 -7 **17**

{1(-2)= -2; 9 + -2 = 7; 7(-2) = -14, and so on...}
Place the variables next to each coefficient starting with one LESS than the highest exponent in the original expression and sequentially lowering until you reach the constant. The last sum is equal to the remainder:

x^2 + 7x - 7 [17/(x+2)]

Well, I hoped I helped you out with your work. Best of luck to you!!