5.5-bo'lim : qisman kasrlar
2. Quyidagi ifodalarning har birining qisman kasr parchalanishini aniqlang.
\[\frac{{34 - 12x}}{{3{x^2} - 10x - 8}}\]Yechim
Birinchi qadam qisman fraksiyaning parchalanish shaklini aniqlashdir. Biroq, buni amalga oshirish uchun, avvalo, maxrajni imkon qadar ko'proq faktorlarga ajratishimiz kerak. Shunday qilib,
\[\frac{{34 - 12x}}{{\left( {3x + 2} \right)\left( {x - 4} \right)}}\]OK, endi biz qisman kasr parchalanishini ko'rishimiz mumkin,
\[\frac{{34 - 12x}}{{3{x^2} - 10x - 8}} = \frac{A}{{3x + 2}} + \frac{B}{{x - 4}}\]2-qadam
Ushbu ifoda uchun LCD displey \(\left( {3x + 2} \right)\left( {x - 4} \right)\). Ikki shartni qo'shish zaxira nusxasini beradi,
\[\frac{{34 - 12x}}{{3{x^2} - 10x - 8}} = \frac{{A\left( {x - 4} \right) + B\left( {3x + 2} \right)}}{{\left( {3x + 2} \right)\left( {x - 4} \right)}}\]3-qadam
Numeratorlarni teng belgilash,
\[34 - 12x = A\left( {x - 4} \right) + B\left( {3x + 2} \right)\]4-qadam
Endi biz konstantalarni aniqlash uchun \(x\) ning "yaxshi" qiymatlarini tanlashimiz kerak. Mana o'sha ish.
\[\begin{array}{l}{x = 4:}\\{\displaystyle x = - \frac{2}{3}:}\end{array}\hspace{0.25in}\begin{aligned} - 14 & = 14B\\42 & = - \frac{{14}}{3}A\end{aligned}\hspace{0.25in} \to \hspace{0.25in}\begin{array}{l}{A = - 9}\\{B = - 1}\end{array}\]5-qadam
Bunda qisman fraksiya parchalanishi,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{34 - 12x}}{{3{x^2} - 10x - 8}} = - \frac{9}{{3x + 2}} - \frac{1}{{x - 4}}}}\]