ঘনফলের সূত্রাবলি ও অনুসিদ্ধান্ত (৪.২)

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সুত্র ৫।

$(a + b)^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

$= a^{3} + b^{3} + 3 a b (a + b)$

প্রমাণ : $(a + b)^{3} = (a + b) (a + b)^{2}$

$= (a + b) (a^{2} + 2 a b + b^{2})$

$= a (a^{2} + 2 a b + b^{2}) + b (a^{2} + 2 a b + b^{2})$

$= a^{3} + 2 a^{2} b + a b^{2} + (a^{2} b + 2 a b^{2} + b^{3})$

$= a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

$= a^{3} + 3 a b (a + b) + b^{3}$

$= a^{3} + b^{3} + 3 a b (a + b)$

অনুসিদ্ধান্ত ৭। $a^{3} + b^{3} = (a + b)^{3} - 3 a b (a + b)$

সুত্র ৬। $(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$

$= a^{3} - b^{3} - 3 a b (a - b)$

প্রমাণ : $(a - b)^{3} = (a - b) (a - b)^{2}$

$= (a - b) (a^{2} - 2 a b + b^{2})$

$= a (a^{2} - 2 a b + b^{2}) - b (a^{2} - 2 a b + b^{2})$

$= a^{3} - 2 a^{2} b + a b^{2} - a^{2} b + 2 a b^{2} - b^{3}$

$= a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}$

$= a^{3} - b^{3} - 3 a b (a - b)$

অনুসিদ্ধান্ত ৮। $a^{3} - b^{3} = (a - b)^{3} + 3 a b (a - b)$

উদাহরণ ১৬। $3 x + 2 y$ এর ঘন নির্ণয় কর।

সমাধান : $(3 x + 2 y)^{3} = (3 x)^{3} + 3 \times (3 x)^{2} \times (2 y) + 3 (3 x) \times (2 y)^{2} + (2 y)^{3}$

$= 27 x^{3} + 3 \times 9 x^{2} \times 2 y + 3 + 3 x \times 4 y^{2} + 8 y^{3}$

$= 27 x^{3} + 54 x^{2} y + 36 x y^{2} + 8 y^{3}$

উদাহরণ ১৭। $2 a + 5 b$ এর ঘন নির্ণয় কর।

সমাধান : $(a + 5 b)^{3} = (2 a)^{3} + 3 \times (2 a)^{2} \times (5 b) + 3 (2 a) \times (5 b)^{2} + (5 b) ³$

$= 8 a^{3} + 3 \times 4 a^{2} \times 5 b + 3 \times 2 a \times 25 b^{2} + 125 b^{3}$

$= 8 a^{3} + 60 a^{2} b + 150 a b^{2} + 125 b^{3}$

উদাহরণ ১৮। $1 m - 2 n$ এর ঘন নির্ণয় কর।

সমাধান : ${(m 2)}^{3} = (m) ³ - 3 \times (m) ² \times (2 n) + 3 \times m \times (2 n) ² - (2 n) ³$

$= m^{3} - 3 m^{2} \times 2 n + 3 m \times 4 n^{2} - 8 n^{3}$

$= m^{3} - 6 m^{2} n + 12 m n^{2} - 8 n^{3}$

উদাহরণ ১৯। $4 x - 5 y$ এর ঘন নির্ণয় কর।

সমাধান : $(4 x - 5 y)^{3} = (4 x)^{3} - 3 \times (4 x)^{2} \times (5 y) + 3 (4 x) \times (5 y)^{2} - (5 y)^{3}$

$= 64 x^{3} - 3 \times 16 x^{2} \times 5 y + 3 \times 4 x \times 25 y^{2} - 125 y^{3}$

$= 64 x^{3} - 240 x^{2} y + 300 x y^{2} - 125 y^{3}$

উদাহরণ ২০। $x + y - z$ এর ঘন নির্ণয় কর।

সমাধান : $(x + y - z)^{3} = {(x + y) - z}^{3}$

$= (x + y)^{3} - 3 (x + y)^{2} z + 3 (x + y) z^{2} - z^{3}$

$= (x^{3} + 3 x^{2} y + 3 x y^{2} + y^{3}) - 3 (x^{2} + 2 x y + y^{2}) \times z + 3 (x + y) \times z^{2} - z^{2}$

$= x ² + 3 x ² y + 3 x y ² + y ³ - 3 x ² z - 6 x y z - 3 y ² z + 3 x z ² + 3 y z ² - z ³$

$= x ³ + y ³ - z ³ + 3 x ² y + 3 x y ² - 3 x ² z - 3 y ² z + 3 x z ² + 3 y z ² - 6 x y z$

কাজ : সূত্রের সাহায্যে ঘন নির্ণয় কর : ১। $a b + b c$ ২। $2 x - 5 y$ ৩। $2 x - 3 y - z$

উদাহরণ ২১। সরল কর : $(4 m + 2 n) ³ + 3 (4 m + 2 n) ² (m - 2 n) + 3 (4 m + 2 n) (m - 2 n) ² + (m - 2 n) ³$

সমাধান : ধরি, $4 m + 2 n = a$ এবং $m - 2 n = b$

$∴$ প্রদত্ত রাশি $= a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}$

$= {(a + b)}^{3}$

$= {(4 m + 2 n) + (m - 2 n)}^{3}$

$= {(4 m + 2 n + m - 2 n)}^{3}$

$= {(5 m)}^{3} = 125 m^{3}$

উদাহরণ ২২। সরল কর : ${(4 a - 8 b)}^{3} - {(3 a - 9 b)}^{3} — 3 (a + b) (4 a - 8 b) (3 a - 9 b)$

সমাধান : ধরি, $4 a - 8 b = x$ এবং $3 a - 9 b = y$

$∴$ $x - y = (4 a - 8 b) - (3 a - 9 b) = 4 a - 8 b - 3 a + 9 b = a + b$

এখন প্রদত্ত রাশি $= x - y^{3} - 3 (x - y) \times x \times y$

$= x^{3} - y^{3} - 3 x y (x - y)$

$= {(x - y)}^{3}$

$= {(a + b)}^{3}$

উদাহরণ ২৩। $a + b = 3$ এবং $a b = 2$ হলে, $a^{3} + b^{3}$ এর মান নির্ণয় কর।

সমাধান : $a^{3} + b^{3} = {(a + b)}^{3} - 3 a b (a + b)$

$= {(3)}^{3} - 3 \times 2 \times 3$

$= 27 - 18$

$= 9$

বিকল্প সমাধান : দেওয়া আছে, $a + b + c$ এবং $a b = 2$

এখন, $a + b + c$

বা, $(a + b)^{3} = (3)^{3}$

বা, $a^{3} + b^{3} + 3 a b (a + b) = 27$

বা, $a^{3} + b^{3} + 3 \times 2 \times 3 = 27$

বা, $a^{3} + b^{3} + 18 = 27$

বা, $a^{3} + b^{3} = 27 - 18$

$∴$ $a^{3} + b^{3} = 9$

উদাহরণ ২৪। $x - y = 10$ এবং $x y = 30$ হলে, $x^{3} - y^{3}$ এর মান নির্ণয় কর।

সমাধান : $x^{3} - y^{3} = (x - y)^{3} + 3 x y (x - y)$

$= (10)^{3} + 3 \times 30 \times 10$

$= 1000 + 900$

$= 1900$

উদাহরণ ২৫। $x + y = 4$ হলে $x^{3} + y^{3} + 12 x y$ এর মান কত?

সমাধান : $x^{3} + y^{3} + 12 x y = x^{3} + y^{3} + 3 \times 4 \times x y$

$= x^{3} + y^{3} + 3 (x + y) \times x y$

$= x^{3} + y^{3} + 3 x y (x + y)$

$= (x + y)^{3}$

$= (4)^{3}$

$= 64 .$

উদাহরণ ২৬। $a + \frac{1}{a} = 7$ হলে, $a^{3} + \frac{1}{a^{3}}$ এর মান নির্ণয় কর।

সমাধান : $a^{3} + \frac{1}{a^{3}} = a^{3} + {(\frac{1}{a})}^{3}$

$= {(a + \frac{1}{a})}^{3} - 3 \times a \times \frac{1}{a} (a + \frac{1}{a})$

$= {(7)}^{3} - 3 \times 7$

$= 343 - 21$

$= 322$

উদাহরণ ২৭। $m = 2$ হলে, $27 m^{3} + 54 m^{2} + 36 m + 3$ এর মান নির্ণয় কর।

সমাধান : প্রদত্ত রাশি $= 27 m^{3} + 54 m^{2} + 36 m + 3$

$= (3 m)^{3} + 3 \times (3 m)^{2} \times 2 + 3 \times (3 m) \times (2)^{2} + (2)^{3} - 5$

$= (3 m + 2)^{3} - 5$

$= (3 \times 2 + 2)^{3} - 5$ [m এর মান বসিয়ে]

$= (6 + 2)^{3} - 5 = 8^{3} - 5$

কাজ : ১। সরল কর : $(7 x - 6)^{3} - (5 x - 6)^{3} - 6 x (7 x - 6) (5 x - 6)$ ২। $a + b = 10$ এবং $a b = 21$ হলে, $a^{3} + b^{3}$ এর মান নির্ণয় কর। ৩। $a + \frac{1}{a} = 3$ হলে, দেখাও যে, $a^{3} + \frac{1}{a^{3}} = 18$

common.content_added_and_updated_by

Rezwan Siddiki Tamim

বীজগণিতীয় সূত্রাবলি অনুশীলনী ৪.১ ঘনফলের সাথে সম্পৃক্ত আরও দুইটি সূত্র অনুশীলনী ৪.২ উৎপাদকে বিশ্লেষণ

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Backtrace	vendor/kalnoy/nestedset/src/QueryBuilder.php:38 vendor/kalnoy/nestedset/src/QueryBuilder.php:60 vendor/kalnoy/nestedset/src/QueryBuilder.php:235 vendor/kalnoy/nestedset/src/QueryBuilder.php:227 vendor/kalnoy/nestedset/src/QueryBuilder.php:305

Bindings	0: 39615 1: 39812
Backtrace	vendor/kalnoy/nestedset/src/QueryBuilder.php:305 vendor/kalnoy/nestedset/src/QueryBuilder.php:321 app/Repositories/SubjectRepository.php:293 vendor/laravel/framework/src/Illuminate/Cache/Repository.php:427 vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:453

Bindings	0: App\Models\Subject 1: 14010
Backtrace	app/Repositories/SubjectRepository.php:325 app/Services/SubjectService.php:76 app/Http/Controllers/Academy/SubjectController.php:105 vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54 vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44

Bindings	0: 0 1: youtube
Backtrace	app/Repositories/SubjectRepository.php:340 app/Services/SubjectService.php:77 app/Http/Controllers/Academy/SubjectController.php:105 vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54 vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44

Bindings	0: 14010 1: 7875 2: 2
Backtrace	app/Services/SubjectService.php:82 app/Http/Controllers/Academy/SubjectController.php:105 vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54 vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:44 vendor/laravel/framework/src/Illuminate/Routing/Route.php:266

Bindings	0: active
Backtrace	view::includes.restrict:30 vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124 vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58 vendor/livewire/livewire/src/Mechanisms/ExtendBlade/ExtendedCompilerEngine.php:22 vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:75

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