5.5-bo'lim : qisman kasrlar
8. Quyidagi ifodalarning har birining qisman kasr parchalanishini aniqlang.
\[\frac{{3{x^2} + 7x + 28}}{{x\left( {{x^2} + x + 7} \right)}}\]Yechim
Birinchi qadam qisman fraksiyaning parchalanish shaklini aniqlashdir. Ushbu muammo uchun qisman kasr parchalanishi,
\[\frac{{3{x^2} + 7x + 28}}{{x\left( {{x^2} + x + 7} \right)}} = \frac{A}{x} + \frac{{Bx + C}}{{{x^2} + x + 7}}\]2-qadam
Ushbu ifoda uchun LCD displey \(x\left( {{x^2} + x + 7} \right)\). Shartlarni qo'shish zaxira nusxasini beradi,
\[\frac{{3{x^2} + 7x + 28}}{{x\left( {{x^2} + x + 7} \right)}} = \frac{{A\left( {{x^2} + x + 7} \right) + \left( {Bx + C} \right)\left( x \right)}}{{x\left( {{x^2} + x + 7} \right)}}\]3-qadam
Numeratorlarni teng belgilash,
\[3{x^2} + 7x + 28 = A\left( {{x^2} + x + 7} \right) + \left( {Bx + C} \right)\left( x \right)\]4-qadam
Bizda aniqlab bo'lmaydigan kvadrat tenglama mavjud bo'lganligi sababli, ushbu bo'limning birinchi masalalarida qo'llagan usulimiz ishlamaydi. Shunday qilib, biz hamma narsani ko'paytirishimiz va shunga o'xshash shartlarni to'plashimiz kerak.
\[\begin{align*}3{x^2} + 7x + 28 & = A{x^2} + Ax + 7A + B{x^2} + Cx\\ & = \left( {A + B} \right){x^2} + \left( {A + C} \right)x + 7A\end{align*}\]5-qadam
Endi biz koeffitsientlarni tenglashtirishimiz kerak. Esda tutingki, bu faqat \({x^2}\) koeffitsientini \(x\) koeffitsientlari va konstantalar uchun ikkala tomonga teng va o'xshash tarzda belgilashni anglatadi. Shunday qilib,
\[\begin{align*}A + B & = 3\\ A + C & = 7\\ 7A & = 28\end{align*}\]6-qadam
Izohlarda aytib o'tilganidek, bu tenglamalar sistemasi bo'lib, biz umuman qanday hal qilish haqida gapirmaganmiz, ammo bu haqiqiy muammo emas. Uchinchi tenglamadan biz \(A = 4\) ga ega bo'lishimiz kerakligini ko'rishimiz mumkin.
Buni olganimizdan so'ng, biz \(B\) va \(C\) qiymatlarini aniqlash uchun uni faqat birinchi ikkita tenglamaga ulashimiz kerak. Mana bu ish,
\[\begin{array}{l}{4 + B = 3}\\{4 + C = 7}\end{array}\hspace{0.25in} \to \hspace{0.25in}\begin{array}{l}{B = - 1}\\{C = 3}\end{array}\]7-qadam
Bunda qisman fraksiya parchalanishi,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{3{x^2} + 7x + 28}}{{x\left( {{x^2} + x + 7} \right)}} = \frac{4}{x} + \frac{{3 - x}}{{{x^2} + x + 7}}}}\]