e)(4x^2 - 3x + 17 over x^3 - 1 + 2x - 1 over x^2 + x + 1 + 6 over 1 - x)

Hướng dẫn: Đưa những phân thức về cùng mẫu bằng phương pháp quy đồng rồi cộng tử với nhau.

a)(5 over 2x^2y + 3 over 5xy^2 + x over y^3)

(= 5.5y^2 over 2x^2y.5y^2 + 3.2xy over 5xy^2.2xy + x.10x^2 over y^3.10x^2)

( = 25y^2 over 10x^2y^3 + 6xy over 10x^2y^3 + 10x^3 over 10x^2y^3)

(= 25y^2 + 6xy + 10x^3 over 10x^2y^3)

b)(x + 1 over 2x + 6 + 2x + 3 over xleft( x + 3 ight))

(= x + 1 over 2left( x + 3 ight) + 2x + 3 over xleft( x + 3 ight))

( = xleft( x + 1 ight) over 2xleft( x + 3 ight) + 2left( 2x + 3 ight) over 2xleft( x + 3 ight))

(= x^2 + x + 4x + 6 over 2xleft( x + 3 ight))

( = x^2 + 5x + 6 over 2xleft( x + 3 ight))

(= x^2 + 2x + 3x + 6 over 2xleft( x + 3 ight))

(= xleft( x + 2 ight) + 3left( x + 2 ight) over 2xleft( x + 3 ight))

c)(3x + 5 over x^2 - 5x + 25 - x over 25 - 5x)

( = 3x + 5 over x^2 - 5x + x - 25 over 5x - 25)

( = 3x + 5 over xleft( x - 5 ight) + x - 25 over 5left( x - 5 ight))

( = 5left( 3x + 5 ight) over 5xleft( x - 5 ight) + xleft( x - 25 ight) over 5xleft( x - 5 ight))

( = 15x + 25 + x^2 - 25x over 5xleft( x - 5 ight) = x^2 - 10x + 25 over 5xleft( x - 5 ight))

( = left( x - 5 ight)^2 over 5xleft( x - 5 ight) = x - 5 over 5x)

d)(x^2 + x^4 + 1 over 1 - x^2 + 1)

(= 1 + mx^2 + x^4 + 1 over 1 - x^2)

( = left( 1 + x^2 ight)left( 1 - x^2 ight) over 1 - x^2 + x^4 + 1 over 1 - x^2)

(= 1 - x^4 + x^4 + 1 over 1 - x^2 = 2 over 1 - x^2)

e)(4x^2 - 3x + 17 over x^3 - 1 + 2x - 1 over x^2 + x + 1 + 6 over 1 - x)

(4x^2 - 3x + 17 over left( x - 1 ight)left( x^2 + x + 1 ight) + 2x - 1 over x^2 + x + 1 + - 6 over x - 1)

( = 4x^2 - 3x + 17 over left( x - 1 ight)left( x^2 + x + 1 ight) + left( 2x - 1 ight)left( x - 1 ight) over left( x - 1 ight)left( x^2 + x + 1 ight) + - 6left( x^2 + x + 1 ight) over left( x - 1 ight)left( x^2 + x + 1 ight))

( = 4x^2 - 3x + 17 + left( 2x - 1 ight)left( x - 1 ight) - 6left( x^2 + x + 1 ight) over left( x - 1 ight)left( x^2 + x + 1 ight))

( = 4x^2 - 3x + 17 + 2x^2 - 3x + 1 - 6x^2 - 6x - 6 over left( x - 1 ight)left( x^2 + x + 1 ight))

( = - 12x + 12 over left( x - 1 ight)left( x^2 + x + 1 ight) = - 12left( x - 1 ight) over left( x - 1 ight)left( x^2 + x + 1 ight) = - 12 over x^2 + x + 1)