Quy đồng mẫu số các phân thức sau:
a. \(\frac{3x}{x^3-xy^2};\frac{2}{\left(x+y\right)^2};\frac{1}{\left(x-y\right)^2}\)
b. \(\frac{4}{x^2+5x-6};\frac{5x}{x^2+2x-3}\)
Hướng dẫn giải
a. Ta có \(\frac{3x}{x^3-xy^2}=\frac{3x}{x\left(x+y\right)\left(x-y\right)};\frac{2}{\left(x+y\right)^2};\frac{1}{\left(x-y\right)^2}\)
Vậy mẫu số chung là \(x\left(x+y\right)^2\left(x-y\right)^2\)
Ta có \(\frac{3x}{x^3-xy^2}=\frac{3x}{x\left(x+y\right)\left(x-y\right)}=\frac{3x\left(x^2-y^2\right)}{x\left(x+y\right)^2\left(x-y\right)^2}=\frac{3x^3-3xy^2}{x\left(x+y\right)^2\left(x-y\right)^2}\)
\(\frac{2}{\left(x+y\right)^2}=\frac{2x\left(x-y\right)^2}{x\left(x+y\right)^2\left(x-y\right)^2}=\frac{2x^3-4x^2y+2xy^2}{x\left(x+y\right)^2\left(x-y\right)^2}\)
\(\frac{1}{\left(x-y\right)^2}=\frac{x\left(x+y\right)^2}{x\left(x+y\right)^2\left(x-y\right)^2}=\frac{x^3+2x^2y+xy^2}{x\left(x+y\right)^2\left(x-y\right)^2}\)
b. \(\frac{4}{x^2+5x-6}=\frac{4}{\left(x-1\right)\left(x+6\right)};\frac{5x}{x^2+2x-3}=\frac{5}{\left(x-1\right)\left(x+3\right)}\)
Mẫu số chung là: \(\left(x-1\right)\left(x+3\right)\left(x+6\right)\)
Vậy thì \(\frac{4}{x^2+5x-6}=\frac{4\left(x+3\right)}{\left(x-1\right)\left(x+3\right)\left(x+6\right)}=\frac{4x+12}{\left(x-1\right)\left(x+3\right)\left(x+6\right)}\)
\(\frac{5x}{x^2+2x-3}=\frac{5x\left(x+6\right)}{\left(x-1\right)\left(x+3\right)\left(x+6\right)}=\frac{5x^2+30x}{\left(x-1\right)\left(x+3\right)\left(x+6\right)}\)
Học bài
