Explain how to answer. $\frac{x^{4} + 2x^{2}-2x-3}{x^{2}+ 7x+10}$

Question

Divide $$\frac{x^{4} + 2x^{2}-2x-3}{x^{2}+ 7x+10}$$ using polynomial long division. Please show step by step as I am unsure.

Answer 1

Hint:

$ (x^4 + 2x^2-2x-3)-(x^2)(x^2+ 7x+10) = -7 x^3 - 8 x^2 - 2 x - 3$

$ (-7 x^3 - 8 x^2 - 2 x - 3)-(-7x)(x^2+ 7x+10) = \cdots $

Answer 2

$$x^4+2x^2-2x-3=$$ $$=x^4+7x^3+10x^2-7x^3-49x^2-70x+41x^2+287x+410-219x-413=$$ $$=(x^2-7x+41)(x^2+7x+10)-219x-413,$$ which says $$\frac{x^4+2x^2-2x-3}{x^2+7x+10}=x^2-7x+41+\frac{-219x-413}{x^2+7x+10}.$$

Also we can use $x^2+7x+10=(x+2)(x+5)$

and the Polynomial remainder theorem, but it's very ugly.

Answer 3

The answer of your question is using the algorithm that you've known for the Euclidean division of integers (I am saying then division by the box method that you've learned in the school), but applying such algorithm for polynomials (notice that it can be justified since Euclidean algorithm and Euclidean theorem hold for the ring of polynomials with integer coefficients). To understand the answers/comments you need also to combine with the Euclidean theorem $$\text{dividend}=\text{divisor}\cdot\text{quotient}+\text{remainder},$$and the divisition rule saying that you can perform an step of your algorithm (box method) to calculate a partial division when $$\deg(\text{polynomial being a dividend})\geq \deg(\text{polynomial being a divisor}).$$ Here $\deg$ denotes the degree of those polynomials.

Hint:

Try your algorithm, division by boxes, with a toy example, for example with dividend $3x^2+2$ and divisor $x+1$. Then the quotient is $3x-3$ and remainder $5$.

Or more easy examples as dividend $x-3$ and divisor $x-1$, look at the division by boxes, obviously here the quotient is $1$ and the reaminder $-2$. Since the degree of the polynomial $-2$ is less than the degree of our quotient $x-1$ we've finished this second toy example, and it can be written as, using Euclides statement, $$\frac{x-3}{x-1}=\frac{1}{x-1}-\frac{2}{x-1},$$ since we've divided by $x-1$.

I would like to write my explanation with your example and using division by boxes, the problem is that my skills writting tex aren't the best.

---

Notice that if you want to know more about the method that I evoke, if you want, you can to search the key words: division of polynomials, box method. Then you should find different resources where examples as I've evoked were solved explicitly.