Question

which rational expression should be subtracted from
$x {}^{2} + 6x + 8 \div x {8} + 8$
to get
$3 \div x {}^{2} - 2x + 4$

​

Answer 1

Given : (x² + 6x + 8)/(x⁸ + 8)  and 3/(x² - 2x + 4)

To find : which rational expression should be subtracted

Solution:

Let say p/q should be subtracted

from (x² + 6x + 8)/(x⁸ + 8)  to get  3/(x² - 2x + 4)

=> (x² + 6x + 8)/(x⁸ + 8) - p/q =  3/(x² - 2x + 4)

=> p/q =  (x² + 6x + 8)/(x⁸ + 8) - 3/(x² - 2x + 4)

=>  p/q = ((x² + 6x + 8)(x² - 2x + 4) - 3(x⁸ + 8) )/(x⁸ + 8) (x² - 2x + 4)

(x² + 6x + 8)(x² - 2x + 4)

= x⁴ - 2x³ + 4x²  + 6x³ - 12x² + 24x  + 8x² - 16x + 32

= x⁴ + 4x³ + 8x + 32

(x⁸ + 8) (x² - 2x + 4)

= x¹⁰ - 2x⁹ + 4x⁸ + 8x² -16x + 32

=>  p/q =  (x⁴ + 4x³ + 8x + 32  - 3 x⁸ -24) )/(x¹⁰ - 2x⁹ + 4x⁸ + 8x² -16x + 32)

=>  p/q =  ( - 3 x⁸ + x⁴ + 4x³ + 8x + 8 )  /(x¹⁰ - 2x⁹ + 4x⁸ + 8x² -16x + 32)

( - 3 x⁸ + x⁴ + 4x³ + 8x + 8 )  /(x¹⁰ - 2x⁹ + 4x⁸ + 8x² -16x + 32) should be subtracted

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