#1**0 **

When you have a hard time, just expand the equation:

(x^3-2)/ (x^2-2) : (X)(X)(X)-2 / (x)(x)+X

now proceed to eliminate which gives: (X)-2 / X

X/X= 1 so they cancel out

-2/1 = -2

:)

Guest Sep 28, 2018

#2**0 **

Simplify the following:

(x^3 - 2)/(x^2 + x)

**There**** isn't much you can do with it !!**

Factor common terms out of x^2 + x.

Factor x out of x^2 + x:

**(x^3 - 2)/(x (x + 1)), or you can write it like this:x^3/(x^2 + x) - 2/(x^2 + x)**

Guest Sep 28, 2018

#3**+2 **

I am not sure

All I can think to do is division which gave me

\(\frac{x^3-2}{x^2+x}\\ =x-1+\frac{x-2}{x^2+2x} \qquad \text{(This has been edited)}\)

I will leave this to another mathematician.

**Continued from earlier**

**\(\frac{x^3-2}{x^2+x}\\ =x-1+\frac{x-2}{x^2+2x}\\ =x-1+\frac{x-2}{x(x+2)}\\\)**

**consider just the fraction part**

**\(\frac{x-2}{x(x+2)}=\frac{A}{x}+\frac{B}{x+2} \qquad \text{for some real A and B}\\ A(x+2)+Bx\equiv x-2\\ Ax+2A+Bx\equiv x-2\\ (A+B)x+2A\equiv x-2\\ so\\ 2A=-2\\ A=-1\\ A+B=1\\ -1+B=1\\ B=2\\ \frac{x-2}{x(x+2)}=\frac{-1}{x}+\frac{2}{x+2} \)**

**So**

\(\frac{x^3-2}{x^2+x} \\ =x-1+\frac{x-2}{x^2+2x} \\ =x-1+\frac{-1}{x}+\frac{2}{x+2}\\ =x-1-\frac{1}{x}+\frac{2}{x+2}\\\)

I think that is right now.

What do you think Mathbum and guest?

Melody
Sep 28, 2018

#10**0 **

LOL this must not be my question! it is jinxed

Oh well doesn't matter, the technique is the same.

Mathbum can copy the technique and get the original answer correct :)

Melody
Sep 29, 2018

#11**0 **

At the consider just the fraction part Melody, why would you have x-2 in the numerator on the left side?

mathbum
Sep 29, 2018

#13**+2 **

Becasue I did the algebraic division first (which I did not display) and x-2 was the remiander.

\((x^2+x) \text{ divided into } (x^3-2) \;\;\;goes \;\;\; (x-1) \text{ times with a remainder of } (x-2)\)

\(so\\ \frac{x^3-2}{x^2+x}=x-1+\frac{x-2}{x^2+x} \)

Maybe this will help

\(7 \div 2=3\;\;remainder 1\\ so\\ \frac{7}{2}=3+\frac{1}{2}\)

I hope I have not made any typing errors this time.

Do you know how to do the algebraic division?

Melody
Sep 29, 2018

#4**0 **

So basically you just have to factor out the common factors on both the denominator or numberstor then cancel them out.

hope this helps!

HelpPls
Sep 28, 2018

#12**+2 **

(x^3 - 2) / (x^2 + x)

Partial faction decomposition can only be done whenever the degree of the polynomial in the numerator is < the degree of the polynomial in the denominator

Performing polynomial division, we have

x - 1

x^2 + x [ x^2 + 0x^2 + 0x - 2 ]

x^3 + x^2

___________________

-x^2 + 0x

-x^2 - 1x

_________

x - 2

The result is ( x - 1) + ( x - 2) / ( x^2 + x)

We can decompose the remainder as

( x - 2) / [ x ( x + 1) = A / x + B / ( x + 1)

Multiply through by x ( x + 1)

x - 2 = A(x + 1) + Bx simplify

x - 2 = (A + B)x + A equate coefficients

A + B = 1

A = -2

Which means that B = 3

So...the complete decomposition is

(x -1 ) - [ 2 / x ] + [ 3 / ( x + 1) ]

CPhill
Sep 29, 2018