(2+1/x)^2/1+x-(1-1/x)^2*(x+1/x-2)-frac{2x+1}{x^2+x} eching

$$\frac { ( 2 + \frac { 1 } { x } ) ^ { 2 } } { 1 + x } - ( 1 - \frac { 1 } { x } ) ^ { 2 } \cdot ( x + \frac { 1 } { x } - 2 ) - \frac { 2 x + 1 } { x ^ { 2 } + x }$$

Baholash

$$-x+4-\frac{4}{x}+\frac{5}{x^{2}}-\frac{1}{x^{3}}$$

Qisqa yechim qadamlari

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2 ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{\left(\frac{2x}{x}+\frac{1}{x}\right)^{2}}{1+x}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{2x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(\frac{2x+1}{x}\right)^{2}}{1+x}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{2x+1}{x}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\frac{\left(2x+1\right)^{2}}{x^{2}}}{1+x}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{\frac{\left(2x+1\right)^{2}}{x^{2}}}{1+x} ni yagona kasrga aylantiring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\left(\frac{x}{x}-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\left(\frac{x-1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{x-1}{x}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x-2 ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\left(\frac{\left(x-2\right)x}{x}+\frac{1}{x}\right)-\frac{2x+1}{x^{2}+x}$$

\frac{\left(x-2\right)x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\times \frac{\left(x-2\right)x+1}{x}-\frac{2x+1}{x^{2}+x}$$

\left(x-2\right)x+1 ichidagi ko‘paytirishlarni bajaring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\times \frac{x^{2}-2x+1}{x}-\frac{2x+1}{x^{2}+x}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{\left(x-1\right)^{2}}{x^{2}} ni \frac{x^{2}-2x+1}{x} ga ko‘paytiring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)}{x^{2}x}-\frac{2x+1}{x^{2}+x}$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 2 va 1 ni qo‘shib, 3 ni oling.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)}{x^{3}}-\frac{2x+1}{x^{2}+x}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x^{2}\left(1+x\right) va x^{3} ning eng kichik umumiy karralisi \left(x+1\right)x^{3}. \frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)} ni \frac{x}{x} marotabaga ko'paytirish. \frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)}{x^{3}} ni \frac{x+1}{x+1} marotabaga ko'paytirish.

$$\frac{\left(2x+1\right)^{2}x}{\left(x+1\right)x^{3}}-\frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right)}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

\frac{\left(2x+1\right)^{2}x}{\left(x+1\right)x^{3}} va \frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right)}{\left(x+1\right)x^{3}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{\left(2x+1\right)^{2}x-\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right)}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

\left(2x+1\right)^{2}x-\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{4x^{3}+4x^{2}+x-x^{5}+x^{4}+x^{3}-x^{2}+2x^{4}-2x^{3}-2x^{2}+2x-x^{3}+x^{2}+x-1}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

4x^{3}+4x^{2}+x-x^{5}+x^{4}+x^{3}-x^{2}+2x^{4}-2x^{3}-2x^{2}+2x-x^{3}+x^{2}+x-1 kabi iboralarga o‘xshab birlashtiring.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

Faktor: x^{2}+x.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}-\frac{2x+1}{x\left(x+1\right)}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \left(x+1\right)x^{3} va x\left(x+1\right) ning eng kichik umumiy karralisi \left(x+1\right)x^{3}. \frac{2x+1}{x\left(x+1\right)} ni \frac{x^{2}}{x^{2}} marotabaga ko'paytirish.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}-\frac{\left(2x+1\right)x^{2}}{\left(x+1\right)x^{3}}$$

\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}} va \frac{\left(2x+1\right)x^{2}}{\left(x+1\right)x^{3}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-\left(2x+1\right)x^{2}}{\left(x+1\right)x^{3}}$$

2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-\left(2x+1\right)x^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-2x^{3}-x^{2}}{\left(x+1\right)x^{3}}$$

2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-2x^{3}-x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}$$

\frac{x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\frac{\left(x+1\right)\left(-x^{4}+4x^{3}-4x^{2}+5x-1\right)}{\left(x+1\right)x^{3}}$$

Surat va maxrajdagi ikkala x+1 ni qisqartiring.

$$\frac{-x^{4}+4x^{3}-4x^{2}+5x-1}{x^{3}}$$

Kengaytirish

$$-x+4-\frac{4}{x}+\frac{5}{x^{2}}-\frac{1}{x^{3}}$$

Qisqa yechim qadamlari

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2 ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{\left(\frac{2x}{x}+\frac{1}{x}\right)^{2}}{1+x}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{2x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(\frac{2x+1}{x}\right)^{2}}{1+x}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{2x+1}{x}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\frac{\left(2x+1\right)^{2}}{x^{2}}}{1+x}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{\frac{\left(2x+1\right)^{2}}{x^{2}}}{1+x} ni yagona kasrga aylantiring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\left(1-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 1 ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\left(\frac{x}{x}-\frac{1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\left(\frac{x-1}{x}\right)^{2}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

\frac{x-1}{x}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\left(x+\frac{1}{x}-2\right)-\frac{2x+1}{x^{2}+x}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x-2 ni \frac{x}{x} marotabaga ko'paytirish.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\left(\frac{\left(x-2\right)x}{x}+\frac{1}{x}\right)-\frac{2x+1}{x^{2}+x}$$

\frac{\left(x-2\right)x}{x} va \frac{1}{x} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\times \frac{\left(x-2\right)x+1}{x}-\frac{2x+1}{x^{2}+x}$$

\left(x-2\right)x+1 ichidagi ko‘paytirishlarni bajaring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}}{x^{2}}\times \frac{x^{2}-2x+1}{x}-\frac{2x+1}{x^{2}+x}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{\left(x-1\right)^{2}}{x^{2}} ni \frac{x^{2}-2x+1}{x} ga ko‘paytiring.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)}{x^{2}x}-\frac{2x+1}{x^{2}+x}$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 2 va 1 ni qo‘shib, 3 ni oling.

$$\frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)}-\frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)}{x^{3}}-\frac{2x+1}{x^{2}+x}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. x^{2}\left(1+x\right) va x^{3} ning eng kichik umumiy karralisi \left(x+1\right)x^{3}. \frac{\left(2x+1\right)^{2}}{x^{2}\left(1+x\right)} ni \frac{x}{x} marotabaga ko'paytirish. \frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)}{x^{3}} ni \frac{x+1}{x+1} marotabaga ko'paytirish.

$$\frac{\left(2x+1\right)^{2}x}{\left(x+1\right)x^{3}}-\frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right)}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

\frac{\left(2x+1\right)^{2}x}{\left(x+1\right)x^{3}} va \frac{\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right)}{\left(x+1\right)x^{3}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{\left(2x+1\right)^{2}x-\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right)}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

\left(2x+1\right)^{2}x-\left(x-1\right)^{2}\left(x^{2}-2x+1\right)\left(x+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\frac{4x^{3}+4x^{2}+x-x^{5}+x^{4}+x^{3}-x^{2}+2x^{4}-2x^{3}-2x^{2}+2x-x^{3}+x^{2}+x-1}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

4x^{3}+4x^{2}+x-x^{5}+x^{4}+x^{3}-x^{2}+2x^{4}-2x^{3}-2x^{2}+2x-x^{3}+x^{2}+x-1 kabi iboralarga o‘xshab birlashtiring.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}-\frac{2x+1}{x^{2}+x}$$

Faktor: x^{2}+x.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}-\frac{2x+1}{x\left(x+1\right)}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \left(x+1\right)x^{3} va x\left(x+1\right) ning eng kichik umumiy karralisi \left(x+1\right)x^{3}. \frac{2x+1}{x\left(x+1\right)} ni \frac{x^{2}}{x^{2}} marotabaga ko'paytirish.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}-\frac{\left(2x+1\right)x^{2}}{\left(x+1\right)x^{3}}$$

\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}} va \frac{\left(2x+1\right)x^{2}}{\left(x+1\right)x^{3}} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-\left(2x+1\right)x^{2}}{\left(x+1\right)x^{3}}$$

2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-\left(2x+1\right)x^{2} ichidagi ko‘paytirishlarni bajaring.

$$\frac{2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-2x^{3}-x^{2}}{\left(x+1\right)x^{3}}$$

2x^{3}+2x^{2}+4x-x^{5}+3x^{4}-1-2x^{3}-x^{2} kabi iboralarga o‘xshab birlashtiring.

$$\frac{x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}}$$

\frac{x^{2}+4x-x^{5}+3x^{4}-1}{\left(x+1\right)x^{3}} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\frac{\left(x+1\right)\left(-x^{4}+4x^{3}-4x^{2}+5x-1\right)}{\left(x+1\right)x^{3}}$$

Surat va maxrajdagi ikkala x+1 ni qisqartiring.

$$\frac{-x^{4}+4x^{3}-4x^{2}+5x-1}{x^{3}}$$