Rút gọn các phân thức

Bài 2:

a) B = 2x² - 3x - 5 = (2x - 5)(x + 1)

\(\frac{\left(2x-5\right)\left(x+1\right)}{\left(3x^2+4\right)\left(x+1\right)}=\frac{2x^2-3x-5}{3x^3+3x^2+4x+4}\)

b) \(\frac{\left(x+1\right)\left(x^2+x-6\right)}{\left(x^2-9\right)\left(x^2+3x+2\right)}=\frac{\left(x+1\right)\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x-3\right)\left(x+1\right)\left(x+2\right)}=\frac{x-2}{\left(x-3\right)\left(x+2\right)}\)

Bài 1:

a) \(\frac{A}{3x-2}=\frac{15x^2+10x}{9x^2-4}=\frac{5x\left(3x+2\right)}{\left(3x+2\right)\left(3x-2\right)}=\frac{5x}{3x-2}\)

=> \(\frac{A}{3x-2}=\frac{5x}{3x-2}\)

=> A = 5x

b) Ta có: 3x² - 5x -2 = (3x + 1)(x - 2)

=> \(\frac{\left(3x+1\right)\left(x-2\right)}{A}=\frac{x-2}{2x-3}\)

=> A = (3x +1)(2x - 3) = 6x² - 7x - 3

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

=> \(\frac{x+2}{x+3}=\frac{x^2+4x+4}{A}=\frac{\left(x+2\right)^2}{A}\)

=> A = (x + 3)(x + 2) = x² + 5x + 6

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

<=> \(\frac{2x+1}{\left(x+2\right)\left(x^2-x+1\right)}=\frac{A}{\left(x+1\right)\left(x^2-x+1\right)}\)

<=>\(\frac{2x+1}{\left(x+2\right)}=\frac{A}{\left(x+1\right)}\)

=> \(\frac{\left(2x+1\right)\left(x+1\right)}{\left(x+2\right)}=A\)

Bài 3:

a) \(\frac{2^{18}.54^3+15.4^{10}.9^4}{2.12^9+6^{10}.2^{10}}=\frac{2^{18}.\left(2.3^3\right)^3+15.2^{20}.3^8}{2.\left(2^2.3\right)^3+2^{10}.3^{10}.2^{10}}\)

\(=\frac{2^{21}.3^9+15.2^{20}.3^8}{2^{19}.3^9+2^{20}.3^{10}}\)

\(=\frac{2^{20}.3^8.\left(3.2\ +15\right)}{2^{19}.3^9.\left(1+2.3\right)}=\frac{2.21}{3.7}=2\)