1.6-bo'lim : Ratsional ifodalar
5. Quyidagi ifodada ko'rsatilgan amalni bajaring va javobni eng kichik shartlarga qisqartiring.
\[\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}}\]Yechim
Shunday qilib, biz birinchi navbatda buni mahsulotga aylantirishimiz kerak.
\[\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}} = \frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}}\,\centerdot \,\frac{{{x^2} + 7x + 6}}{{{x^2} - x - 42}}\]Bo'linishni mahsulotga aylantirmaguningizcha ko'paytuvchilarga ajratish va bekor qilmasligingizga ishonch hosil qiling.
2-qadam
Endi biz shartlarning har birini olish uchun iloji boricha ko'proq faktor bo'lishimiz mumkin,
\[\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}} = \frac{{\left( {x - 7} \right)\left( {x + 7} \right)}}{{\left( {2x - 5} \right)\left( {x + 1} \right)}}\centerdot \frac{{\left( {x + 1} \right)\left( {x + 6} \right)}}{{\left( {x - 7} \right)\left( {x + 6} \right)}}\]3-qadam
Nihoyat, eng past shartlarga qisqartirish va mahsulotni bajarish uchun iloji boricha bekor qiling.
\[\frac{{{x^2} - 49}}{{2{x^2} - 3x - 5}} \div \frac{{{x^2} - x - 42}}{{{x^2} + 7x + 6}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{x + 7}}{{2x - 5}}}}\]