(2đ) Rút gọn các phân thức sau: a) \(\frac{{5(x – y) – 3(y – x)}}{{10(x – y)}}\) b) \(\frac{{{x^2} + 4x + 3}}{{2x + 6}}\) c) \(\frac{{{y^2} – {x^2}}}{{{x^2} – 3xy + 2{y^2}}}\) d) \(\frac{{2{x^3} – 7{x^2} – 12x + 45}}{{3{x^3} – 19{x^2} + 33x – 9}}\)

(2đ) Rút gọn các phân thức sau:

a) \(\frac{{5(x – y) – 3(y – x)}}{{10(x – y)}}\)

b) \(\frac{{{x^2} + 4x + 3}}{{2x + 6}}\)

c) \(\frac{{{y^2} – {x^2}}}{{{x^2} – 3xy + 2{y^2}}}\)

d) \(\frac{{2{x^3} – 7{x^2} – 12x + 45}}{{3{x^3} – 19{x^2} + 33x – 9}}\)

Lời giải chi tiết:

a) \(\frac{{5(x – y) – 3(y – x)}}{{10(x – y)}} = \frac{{5(x – y) + 3(x – y)}}{{10(x – y)}} = \frac{{8(x – y)}}{{10(x – y)}} = \frac{4}{5}\)

b) \(\frac{{{x^2} + 4x + 3}}{{2x + 6}} = \frac{{{x^2} + x + 3x + 3}}{{2(x + 3)}} = \frac{{x(x + 1) + 3(x + 1)}}{{2(x + 3)}} = \frac{{(x + 1)(x + 3)}}{{2(x + 3)}} = \frac{{x + 1}}{2}\) .

c) \(\frac{{{y^2} – {x^2}}}{{{x^2} – 3xy + 2{y^2}}} = \frac{{(y – x)(y + x)}}{{{x^2} – xy – 2xy + 2{y^2}}} = \frac{{(x + y)(y – x)}}{{x(x – y) – 2y(x – y)}} = \frac{{ – (x + y)(x – y)}}{{(x – y)(x – 2y)}} = \frac{{ – (x + y)}}{{x – 2y}}.\)

d) \(\frac{{2{x^3} – 7{x^2} – 12x + 45}}{{3{x^3} – 19{x^2} + 33x – 9}} = \frac{{2{x^3} + 5{x^2} – 12{x^2} – 30x + 18x + 45}}{{3{x^3} – {x^2} – 18{x^2} + 6x + 27x – 9}} = \frac{{{x^2}(2x + 5) – 6x(2x + 5) + 9(2x + 5)}}{{{x^2}(3x – 1) – 6x(3x – 1) + 9(3x – 1)}}\)

\( = \frac{{(2x + 5)({x^2} – 6x + 9)}}{{(3x – 1)({x^2} – 6x + 9)}} = \frac{{2x + 5}}{{3x – 1}}.\)