1.6-bo'lim : Ratsional ifodalar
10. Quyidagi ifodada ko'rsatilgan amalni bajaring.
\[\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}}\]Yechim
Bu ratsional ifoda uchun birinchi navbatda eng kichik umumiy maxraj kerak. Biroq, buni olishdan oldin biz birinchi atamaning maxrajini hisobga olishimiz kerak. Shunday qilib,
\[\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}} = \frac{x}{{{{\left( {x + 6} \right)}^2}}} - \frac{{x - 8}}{{x + 6}}\]2-qadam
Eng kichik umumiy maxraj shundan iboratki,
\[{\mbox{lcd : }}{\left( {x + 6} \right)^2}\]Esda tutingki, biz eng kichik umumiy maxrajni o'rnatishda faqat maxrajdagi har bir atama uchun eng yuqori quvvatni olamiz.
3-qadam
Keyinchalik, har bir atamaning maxrajiga eng kichik umumiy maxrajni olish uchun har bir shartni tegishli miqdorga ko'paytiring.
\[\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}} = \frac{x}{{{{\left( {x + 6} \right)}^2}}} - \frac{{\left( {x - 8} \right)\left( {x + 6} \right)}}{{\left( {x + 6} \right)\left( {x + 6} \right)}}\]4-qadam
Nihoyat, biz qilishimiz kerak bo'lgan narsa ayirish va natijaning numeratorini soddalashtirishdir.
\[\frac{x}{{{x^2} + 12x + 36}} - \frac{{x - 8}}{{x + 6}} = \frac{{x - \left( {x - 8} \right)\left( {x + 6} \right)}}{{{{\left( {x + 6} \right)}^2}}} = \frac{{x - \left( {{x^2} - 2x - 48} \right)}}{{{{\left( {x + 6} \right)}^2}}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{48 + 3x - {x^2}}}{{{{\left( {x + 6} \right)}^2}}}}}\]