How do you divide (3x^4 + 2x^3 - 11x^2 - 2x + 5)/ (x^2 - 2)  using polynomial long division?

Nov 14, 2017

$\frac{3 {x}^{4} + 2 {x}^{3} - 11 {x}^{2} - 2 x + 5}{{x}^{2} - 2} = 3 {x}^{2} + 2 x - 5 + \frac{2 x - 5}{{x}^{2} - 2}$

Explanation:

$3 {x}^{4} + 2 {x}^{3} - 11 {x}^{2} - 2 x + 5 | {x}^{2} - 2$
you ask what is it $\frac{3 {x}^{4}}{x} ^ 2$ and you get $3 {x}^{2}$

(REMEMBER $3 {x}^{2}$)

now yow duplicate $3 {x}^{2}$ to ${x}^{2} - 2$ and get $3 {x}^{4} - 6 {x}^{2}$
and do:
$3 {x}^{4} + 2 {x}^{3} - 11 {x}^{2} - 2 x + 5$
$-$
$3 {x}^{4} + 0 {x}^{3} - 6 {x}^{2} + 0 x + 0$
$=$
$0 + 2 {x}^{3} - 5 {x}^{2} - 2 x + 5$
so we have now $2 {x}^{3} - 5 {x}^{2} - 2 x + 5$ which is very exiting because we don't have the power of 4 anymore!

now we do the same again, try to follow:

First step (write it down):
$2 {x}^{3} - 5 {x}^{2} - 2 x + 5 | {x}^{2} - 2$

Second step (divided strongest power):
$\frac{2 {x}^{3}}{{x}^{2}} = 2 x$

(REMEMBER $2 x$)

Third step (Second step times the divider):
$2 x \cdot \left({x}^{2} - 2\right) = 2 {x}^{3} - 4 x$

Forth step (which is first step minus third step):
$2 {x}^{3} - 5 {x}^{2} - 2 x + 5$
$-$
$2 {x}^{3} + 0 {x}^{2} - 4 x + 0$
$=$
$0 - 5 {x}^{2} + 2 x + 5$

--and agian--

First step (write it down):
$- 5 {x}^{2} + 2 x + 5 | {x}^{2} - 2$

Second step (divided strongest power):
$\frac{- 5 {x}^{2}}{{x}^{2}} = - 5$

(REMEMBER $- 5$)

Third step (Second step times the divider):
$- 5 \cdot \left({x}^{2} - 2\right) = - 5 {x}^{2} + 10$

Forth step (which is first step minus third step):
$- 5 {x}^{2} + 2 x + 5$
$-$
$- 5 {x}^{2} + 10$
$=$
$0 + 2 x - 5$

All REMEMBERs are the result:
$3 {x}^{2} + 2 x - 5$

BUT in that case, $2 x - 5$ could not be divided so it remains as $\frac{2 x - 5}{{x}^{2} - 2}$

SOOOOOOOOOOOOOOOOOOOOOOO :)

$\frac{3 {x}^{4} + 2 {x}^{3} - 11 {x}^{2} - 2 x + 5}{{x}^{2} - 2} = 3 {x}^{2} + 2 x - 5 + \frac{2 x - 5}{{x}^{2} - 2}$

Lets check:

$\left(3 {x}^{2} + 2 x - 5 + \frac{2 x - 5}{{x}^{2} - 2}\right) \left({x}^{2} - 2\right) =$
$= 3 {x}^{4} - 6 {x}^{2} + 2 {x}^{3} - 4 x - 5 {x}^{2} + 10 + 2 x - 5 =$
$= 3 {x}^{4} + 2 {x}^{3} - 11 {x}^{2} - 2 x + 5$

So it's fine :)