ax²+bx+cআকারের রাশির উৎপাদক (৪.৬)

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মনে করি,

${ax}^{2} + bx + c = (x + p) (sx + 9)$

$= {rsx}^{2} + (rq + sp) x + pq$

তাহলে, $a = rs, b = rq + sp$ এবং $c = pq$

সুতরাং, $ac = rspq = rq \times sp$ এবং $b = rq + sp$

এখন, $a x^{2} + b x + c$ আকারের রাশিকে উৎপাদকে বিশ্লেষণ করতে হলে, $x^{2}$ এর সহগ এবং পদ ধ্রুবক $c -$ এর গুণফলকে এমন দুইটি উৎপাদকে প্রকাশ করতে হবে, যেন এদের বীজগণিতীয় যোগফল $x$ এর সহগ $b$ এর সমান হয় এবং $a$ ও $c$ এর গুণফলের সমান হয়।

$2 x^{2} + 11 x + 15$ রাশিটিকে উৎপাদকে বিশ্লেষণ করতে হলে, $(2 \times 15) = 30$ কে এমন দুইটি উৎপাদকে প্রকাশ করতে হবে, যার যোগফল $11$ এবং গুণফল $30$

$30$ এর উৎপাদক জোড়াসমূহ $1, 30, 2, 15, 3, 10$ ও $5, 6$ এর মধ্যে $5, 6$ জোড়াটির যোগফল $5 + 6 = 11$ এবং গুণফল $5 \times 6 = 30$

$∴ 2 x^{2} + 11 x + 15 = 2 x^{2} + 5 x + 6 x + 15$

$= x (2 x + 5) + 3 (2 x + 5) = (2 x + 5) (x + 3)$

মন্তব্য : ${ax}^{2} + bx + c$ এর উৎপাদকে বিশ্লেষণের সময় $x^{2} + p x + q$ এর $p, q$ এর ধনাত্মক ও ঋণাত্মক বিভিন্ন চিহ্নযুক্ত মানের জন্য যে নিয়ম অনুসরণ করা হয়েছে $a, b, c$ এর চিহ্নযুক্ত মানের জন্য একই নিয়ম অনুসরণ করতে হবে। এক্ষেত্রে $p$ এর পরিবর্তে $b$ এবং $q$ এর পরিবর্তে $(a \times c)$ ধরতে হবে।

উদাহরণ ৭। $2 x^{2} + 9 x + 10$ কে উৎপাদকে বিশ্লেষণ কর।

সমাধান : এখানে, $2 \times 10 = 20$ [ $x^{2}$ এর সহগ ও ধ্রুবক পদের গুণফল]

এখন, $4 \times 5 = 20$ এবং $4 + 5 = 9$

$∴ 2 x^{2} + 9 x + 10 = 2 x^{2} + 4 x + 5 x + 10$

$= 2 x (x + 2) + 5 (x + 2) = (x + 2) (2 x + 5)$

উদাহরণ ৮। $3 x^{2} + x - 10$ কে উৎপাদকে বিশ্লেষণ কর।

সমাধান : এখানে, $3 \times (- 10) = 30$

এখন, $(- 5) \times 6 = 30$

$∴ 3 x^{2} + x - 10 = 3 x^{3} + 6 x - 5 x - 10$

$= 3 x (x + 2) - 5 (x + 2)$

$= (x + 2) (3 x - 5)$

উদাহরণ ৯। $4 x^{2} - 23 x + 33$ কে উৎপাদকে বিশ্লেষণ কর।

সমাধান : এখানে, $4 \times 33 = 132$

এখন, $(- 11) \times (- 12) = 132$ এবং $(- 11) + (- 12) = - 23$

$∴ 4 x^{2} - 23 x + 33 = 4 x^{2} - 11 x - 12 x + 33$

$= x (4 x - 11) - 3 (4 x - 11)$

$= (4 x - 11) (x - 3)$

উদাহরণ ১০। $9 x^{2} - 9 x - 4$

সমাধান : এখানে, $9 \times (- 4) = - 36$

এখন, $3 \times (- 12) = 36$ এবং $3 + (- 12) = - 9$

$∴ 9 x^{2} - 9 x - 4 = 9 x^{2} + 3 x - 12 x - 4$

$= 3 x (3 x + 1) - 4 (3 x + 1)$

$= (3 x + 1) (3 x - 4)$

কাজ : উৎপাদকে বিশ্লেষণ কর : $১ । 8 x^{2} + 18 x + 9$ $২ । 27 x^{2} + 15 x + 2$ $৩ । 2 a^{2} - 6 a - 20$

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Rezwan Siddiki Tamim

বীজগণিতীয় সূত্রাবলি অনুশীলনী ৪.১ ঘনফলের সূত্রাবলি ও অনুসিদ্ধান্ত ঘনফলের সাথে সম্পৃক্ত আরও দুইটি সূত্র অনুশীলনী ৪.২

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ax²+bx+cআকারের রাশির উৎপাদক (৪.৬)

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