বীজগণিতীয় ভগ্নাংশের লগিষ্টকরন যোগ ও বিয়োগঃ
এই অধ্যায়ের পূর্বের অংশঃ
JSC (Class 8) Math BD: অষ্টম শ্রেণি সাধারণ গণিতঃ অনুশীলনী-৫.১ বীজগণিতীয় ভগ্নাংশের লগিষ্টকরন যোগ ও বিয়োগ (1-3) Part 1
৪. বিয়োগফল নির্ণয়
করঃ
a a2
(ক) ----- - -------
x-3
x2-9
সমাধানঃ
a a2
----- - -------
x-3 x2-9
a a2
=----- - --------------
x-3 (x-3)(x+3)
a(x+3)-a2
=--------------
(x-3)(x+3)
ax+3a-a2
=-----------
1 1
(খ) -------- -
-------------
y(x-y) x(x+y)
সমাধানঃ
1 1
-------- - --------
y(x-y) x(x+y)
1.x(x+y)-1.y(x-y)
=-------------------
xy(x-y)(x+y)
x2+xy-xy+y2
=---------------
xy(x2-y2)
x2+y2
=-------------
xy(x2-y2)
x+1 x-1
(গ)--------- -
-----------
1+x+x2 1-x+x2
সমাধানঃ
x+1 x-1
--------- - ---------
1+x+x2 1-x+x2
(x+1)(1-x+x2)-(x-1)( 1+x+x2)
=-------------------------------
(1+x+x2)(1-x+x2)
(x+1)(1-x+x2)-(x-1)( 1+x+x2)
=-------------------------------
{(1+x2)+x}{(1+x2)-x}
(x3+1)-(x3-1)
=---------------
(1+x2)2-x2
x3+1-x3+1
=----------------
1+2x2+x4-x2
2
=------------
1+x2+x4
a2+16b2 a-4b
(ঘ)
------------ - -----------
a2-16b2 a+4b
সমাধানঃ
a2+16b2 a-4b
------------ - ---------
a2-16b2 a+4b
a2+16b2 a-4b
=------------ - ---------
a2-(4b)2 a+4b
a2+16b2 a-4b
=---------------- - ---------
(a-4b)(a+4b) a+4b
a2+16b2-(a-4b)(a-4b)
=-----------------------
(a-4b)(a+4b)
a2+16b2-(a-4b)2
=-------------------
(a-4b)(a+4b)
a2+16b2-a2+8ab-16b2
=-------------------------
(a-4b)(a+4b)
8ab
=----------------
(a-4b)(a+4b)
1 x2-xy+y2
(ঙ)------- -
-------------
x-y x3+y3
সমাধানঃ
1 x2-xy+y2
------- - ----------
x-y x3+y3
x3+y3-(x-y)(
x2-xy+y2)
=-------------------------
(x-y)( x3+y3)
x3+y3-x3+x2y-xy2+x2y-xy2+y3
=-------------------------------
(x-y)( x3+y3)
2x2y-2xy2+2y3
=-----------------
(x-y)(x3+y3)
2y(x2-xy+y2)
=--------------------------
(x-y)( x+y)(x2-xy+y2)
2y
=---------------
(x-y)(x+y)
2y
=--------
x2-y2
৫. সরল করঃ
(ক)
x-y y-z z-x
-----+-----+-------
xy yz zx
সমাধানঃ
x-y y-z z-x
-----+-----+-----
xy yz zx
z(x-y)+x(y-z)+y(z-x)
=-----------------------
xyz
zx-zy+xy-zx+yz-xy
=-----------------------
xyz
0
=------
xyz
=0
(খ)
x-y y-z z-x
------------ + ------------- + ---------------
(x+y)(y+z) (y+z)(z+x) (z+x)(x+y)
সমাধানঃ
x-y y-z z-x
------------ + -------------+------------
(x+y)(y+z) (y+z)(z+x) (z+x)(x+y)
(z+x)(x-y)+(x+y)(y-z)+(y+z)(z-x)
=------------------------------------
(x+y)(y+z)(z+x)
zx+x2-zy-xy+xy+y2-zx-yz+yz+z2-xy-za
=----------------------------------------------
(x+y)(y+z)(z+x)
x2+y2+z2-xy-yz-xz
=---------------------
(x+y)(y+z)(z+x)
(গ)
y x z
----------- + ------------+-----------------
(x-y)(y-z) (z-x)(x-y) (y-z)(x-z)
সমাধানঃ
y x z
----------- + ----------- + -----------
(x-y)(y-z) (z-x)(x-y) (y-z)(x-z)
y(z-x)+x(y-z)-z(x-y)
=-------------------------
(x-y)(y-z)(z-x)
yz-xy+xy-zx-zx+yz
=----------------------
(x-y)(y-z)(z-x)
-2zx+2yz
=------------------
(x-y)(y-z)(z-x)
-2z(x-y)
=------------------
(x-y)(y-z)(z-x)
-2z
=--------------
(y-z)(z-x)
(ঘ)
1 1 2x
------- + ------ +-----------
x+3y x-3y x2-9y2
সমাধানঃ
1 1 2x
------- + ------ + ---------
x+3y x-3y
x2-9y2
1 1 2x
=-------+------+----------
x+3y x-3y {x2-(3y)2}
1 1 2x
=-------+------+----------------
x+3y x-3y (x-3y)(x+3y)
(x-3y)+(x+3y)+2x
=----------------------
(x-3y)(x+3y)
x-3y+x+3y-2x
=-----------------
(x-3y)(x+3y)
0
=---------------
(x-3y)(x+3y)
=0
(ঙ)
1
2 1 2
----- - ------- + ------ - ----------
x-y 2x+y x+y 2x-y
সমাধানঃ
1 2 1 2
----- - ------- + ------ - -------
x-y 2x+y x+y 2x-y
2x+y-2(x-y) 2x-y-2(x+y)
=------------- + --------------
(x-y)(2x+y) (x+y)(2x-y)
2x+y-2x+2y 2x-y-2x-2y
=------------- + --------------
(x-y)(2x+y) (x+y)(2x-y)
3y -3 y
=------------- + --------------
(x-y)(2x+y) (x+y)(2x-y)
3y(x+y)(2x-y)-3y(x-y)(2x+y)
=----------------------------------
(x-y)(2x+y)(x+y)(2x-y)
3y(2x2+2xy-xy-y2)-3y(2x2-2xy+xy-y2)
=---------------------------------------------
(x2-y2)(4x2-y2)
3y(2x2+xy-y2)-3y(2x2-xy-y2)
=---------------------------------
(x2-y2)(4x2-y2)
6x2y+3xy2-3y3-6x2y+3xy2+3y3
=-------------------------------------
(x2-y2)(4x2-y2)
6xy2
=---------------------
(x2-y2)(4x2-y2)
(চ)
1 x-2 6x
----- -
------------- + ---------
x-2 x2+2x+4 x3+8
সমাধানঃ
1 x-2 6x
----- - ----------- + ---------
x-2 x2+2x+4 x3+8
x2+2x+4-(x-2)(x-2) 6x
=---------------------- + --------
(x-2)(x2+2x+4) x3+8
x2+2x+4-(x2-4x+22) 6x
=---------------------- + --------
(x-2)(x2+2x+4) x3+8
x2+2x+4-x2+4x-4 6x
=---------------------- + --------
(x-2)(x2+2x+4) x3+8
6x 6x
=-------------------- + --------
(x-2)(x2+2x+4) x3+8
[(x-2)(x2+2x+4)=x3+2x2+4x-2x2-4x-8=x3-8]
6x 6x
=--------- + --------
x3-8 x3+8
6x(x3+8)+6x(x3-8)
=---------------------
(x3-8)(x3+8)
6x4+48x+6x4-48x
=--------------------
(x3)2-(8)2
12x4
=----------
(x6-64)
(ছ)
1 1 2 4
----- - ----- - ------ + ---------
x-1 x+1 x2+1 x4+1
সমাধানঃ
1 1 2 4
----- - ----- - ------ + ---------
x-1 x+1 x2+1 x4+1
x+1-(x-1) 2 4
=------------- - ------- + -------
(x-1)(x+1) x2+1 x4+1
x+1-x+1 2 4
=------------- - ------- + -------
(x-1)(x+1) x2+1 x4+1
2 2 4
=-------- - ------- + -------
x2-1 x2+1 x4+1
2(x2+1)-2(x2-1) 4
=------------------ + --------
(x2-1)(x2+1) x4+1
2x2+2-2x2+2 4
=------------------ + --------
(x2-1)(x2+1) x4+1
4 4
=-------- + --------
x4-1 x4+1
4(x4+1)+4 (x4-1)
=-------------------
(x4-1)( x4+1)
4x4+4+4x4-4
=-------------------
(x4-1)( x4+1)
8x4
=-----------
(x8-1)
(জ)
x-y y-z z-x
----------- + ----------
+ ----------
(y-z)(z-x) (z-x)(x-y) (x-y)(x-z)
সমাধানঃ
x-y y-z z-x
----------- + ---------- + -----------
(y-z)(z-x) (z-x)(x-y)
(x-y)(x-z)
x-y y-z z-x
=----------- + ---------- + -----------
(y-z)(z-x) (z-x)(x-y) (x-y)(z-x)
(x-y)2+(y-z)2-(z-x)(y-z)
=-----------------------------
(x-y)(y-z)(z-x)
x2-2xy+y2+y2-2yz+z2-(zy-z2-xy+zx)
=--------------------------------------------
(x-y)(y-z)(z-x)
x2-2xy+y2+y2-2yz+z2-zy+z2+xy-zx
=-----------------------------------------
(x-y)(y-z)(z-x)
x2-xy+2y2-3yz+2z2+xy-zx
=--------------------------------
(x-y)(y-z)(z-x)
(ঝ)
1 1 a
------ + ------- + ------------------
a-b-c a-b+c a2+b2-c2-2ab
সমাধানঃ
এখানে,
a2+b2-c2-2ab
=(a-b)2-c2
=(a-b-c)(a-b+c)
তাহলে,
1 1 a
------+-------+--------------------
a-b-c a-b+c a2+b2-c2-2ab
1 1 a
=------+-------+-------------------
a-b-c a-b+c (a-b-c)(a-b+c)
a-b+c+ a-b-c+a
=-------------------
(a-b-c)(a-b+c)
3a-2b
=----------------
a2+b2-c2-2ab
(ঞ)
1 1
----------------+---------------------
a2+b2-c2+2ab b2+c2-a2+2bc
1
+-------------------
c2+a2-b2+2ca
সমাধানঃ
এখানে,
a2+b2-c2+2ab=(a+b)2-c2=(a+b+c)(a+b-c)
b2+c2-a2+2bc=(b+c)2-a2=(a+b+c)(b+c-a)
c2+a2-b2+2ca=(c+a)2-b2=(a+b+c)(c+a-b)
∴
1 1
----------------+---------------------
a2+b2-c2+2ab b2+c2-a2+2bc
1
+-------------------
c2+a2-b2+2ca
1 1
=------------------+---------------------
(a+b+c)(a+b-c) (a+b+c)(b+c-a)
1
+----------------------
(a+b+c)(c+a-b)
(b+c-a)(c+a-b)+(a+b-c)(c+a-b)
+(a+b-c)(b+c-a)
=---------------------------------------
(a+b+c)(a+b-c)(b+c-a)(c+a-b)
ab-ca-a2-b2-bc+ab+bc+c2-ca+a2
+ab+ca-ab-b2+bc+ca+bc-c2+bc
+ca-a2+b2+bc-ab-bc-c2+ca
=---------------------------------------
(a+b+c)(a+b-c)(b+c-a)(c+a-b)
2ab+2bc+2ca-a2-b2-c2
=--------------------------------------
(a+b+c)(a+b-c)(b+c-a)(c+a-b)
এই অধ্যায় সহ সকল অধ্যায়ের pdf download লিঙ্ক দেখুনঃ Download Free Book মেনুতে।
