1.6-bo'lim : Ratsional ifodalar
11. Quyidagi ifodada ko'rsatilgan amalni bajaring.
\[\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}}\]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,
\[\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}} = \frac{1}{{\left( {x - 6} \right)\left( {x - 7} \right)}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}}\]2-qadam
Eng kichik umumiy maxraj shundan iboratki,
\[{\mbox{lcd : }}\left( {x - 6} \right)\left( {x - 7} \right)\]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.
\[\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}} = \frac{1}{{\left( {x - 6} \right)\left( {x - 7} \right)}} + \frac{{\left( {x + 1} \right)\left( {x - 7} \right)}}{{\left( {x - 6} \right)\left( {x - 7} \right)}} - \frac{{{x^2}\left( {x - 6} \right)}}{{\left( {x - 7} \right)\left( {x - 6} \right)}}\]4-qadam
Nihoyat, biz qilishimiz kerak bo'lgan yagona narsa - qo'shish va ayirish, keyin natijaning hisobini soddalashtirish.
\[\begin{align*}\frac{1}{{{x^2} - 13x + 42}} + \frac{{x + 1}}{{x - 6}} - \frac{{{x^2}}}{{x - 7}} & = \frac{{1 + \left( {x + 1} \right)\left( {x - 7} \right) - {x^2}\left( {x - 6} \right)}}{{\left( {x - 6} \right)\left( {x - 7} \right)}}\\ & = \frac{{1 + {x^2} - 6x - 7 - {x^3} + 6{x^2}}}{{\left( {x - 6} \right)\left( {x - 7} \right)}} = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{ - {x^3} + 7{x^2} - 6x - 6}}{{\left( {x - 6} \right)\left( {x - 7} \right)}}}}\end{align*}\]