5.5-bo'lim : qisman kasrlar
1. Quyidagi ifodalarning har birining qisman kasr parchalanishini aniqlang.
\[\frac{{17x - 53}}{{{x^2} - 2x - 15}}\]Yechim
Birinchi qadam qisman fraksiyaning parchalanish shaklini aniqlashdir. Biroq, buni amalga oshirish uchun, avvalo, maxrajni imkon qadar ko'proq faktorlarga ajratishimiz kerak. Shunday qilib,
\[\frac{{17x - 53}}{{\left( {x - 5} \right)\left( {x + 3} \right)}}\]OK, endi biz qisman kasr parchalanishini ko'rishimiz mumkin,
\[\frac{{17x - 53}}{{{x^2} - 2x - 15}} = \frac{A}{{x - 5}} + \frac{B}{{x + 3}}\]2-qadam
Ushbu ifoda uchun LCD displey \(\left( {x - 5} \right)\left( {x + 3} \right)\). Ikki shartni qo'shish zaxira nusxasini beradi,
\[\frac{{17x - 53}}{{{x^2} - 2x - 15}} = \frac{{A\left( {x + 3} \right) + B\left( {x - 5} \right)}}{{\left( {x - 5} \right)\left( {x + 3} \right)}}\]3-qadam
Numeratorlarni teng belgilash,
\[17x - 53 = A\left( {x + 3} \right) + B\left( {x - 5} \right)\]4-qadam
Endi biz konstantalarni aniqlash uchun \(x\) ning "yaxshi" qiymatlarini tanlashimiz kerak. Mana o'sha ish.
\[\begin{array}{l}{x = 5:}\\{x = - 3:}\end{array}\hspace{0.25in}\begin{aligned}32 & = 8A\\ - 104 & = - 8B\end{aligned}\hspace{0.25in}\to \hspace{0.25in}\begin{array}{l}{A = 4}\\{B = 13}\end{array}\]5-qadam
Bunda qisman fraksiya parchalanishi,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{17x - 53}}{{{x^2} - 2x - 15}} = \frac{4}{{x - 5}} + \frac{{13}}{{x + 3}}}}\]