Dividing Polynomials: (-12-72*x^3)/(1-2*x-3*x^4) Using Algebra?

Question

Can someone get me started on this? I do not want you to actually solve the equation. Thank you!

If this helps, I am also trying to take the antiderivative of that equation. But I only need to know how to divide those two; I can antidifferentiate on my own.

Answer 1

You don't actually need to do any division in this case - as the above poster mentioned, you can't, because the degree of the top is less than the degree of the bottom.

Thus, when you get stuck like this, either its going to be "impossible" to integrate (at the level you're at, at least), or you're going to get lucky. Which means the top will be a multiple of (the bottom differentiated).
Thats because its easy to integrate f'(x) / f(x) because of the chain rule: the integral of that is ln(f(x)).

In this case, you are lucky (as you always tend to be in textbook questions :)), because the bottom differentiated is -2 - 12x^3, and the top is 6(-2 - 12x^3).
I'll leave the rest to you.

Answer 2

Are you sure you have it correct? The degree of the top (3) is less than the degree of the bottom (4), and when dividing, the degree of the top is almost always greater than or equal to the degree of the bottom.

e.g. (x^2 + 5x + 3) / (x+3) where the degree of the top is 2 and the bottom is 1.

Answer 3

Although not wanting to hurt my brain at the moment, i can tell you the strategy for dividing polynomials, firstly you must "factorise" both top and bottom, then (hopefully) you will be able to cancel out 'like terms", after that further simplify answer with basic algebra skills. Hope this helps.

Kevin.

Answer 4

(-12 - 72x³)
------------------ =
(1 - 2x - 3x^4)

-12 (1 + 6x³)
--------------- ---------------- =
(1 + x)(1 - 3x + 3x² - 3x³)

A ......... B + Cx + Dx²
------- + ---------------------
1 + x ... 1 - 3x + 3x² - 3x³

So A(1 - 3x + 3x² - 3x³) + (B + Cx + Dx²)(1 + x) = -12 (1 + 6x³)

When x = -1 10A = 60 so A = 6

When x = 0 A + B = -12 so B = -18

When x = 1
-6A + 2B + 2C + 2D = -84

ie C + D = 3A - B - 84 = - 48

When x = 2

-17A + 3B + 6C + 12D = -12*49
6C + 12D = -588 + 17 * 6 + 12 * 18 = - 270

So C + 2D = -30
and C + D = -48
Thus D = 18 and C = - 66

Therefore

-12 (1 + 6x³)
---------------- --------------- =
(1 + x)(1 - 3x + 3x² - 3x³)

6 ........ 18 + 66x -18x²
------- - --------------------- =
1 + x .. 1 - 3x + 3x² - 3x³

6 ........ 6(3 + 11x - 3x²)
------- - ---------------------
1 + x .. 1 - 3x + 3x² - 3x³

Answer 5

-12-(72x)^3/1-2x-(3x)^4
=-12-373248x^3/1-2x-81x^4