অনুশীলনী-14(B)
শংকৰদেৱ শিশু নিকেটনৰ অষ্টম শ্ৰেণীৰ গণিত পাঠ্যক্ৰমত উৎপাদক বিশ্লেষণ (Factorization) এক গুৰুত্বপূৰ্ণ অধ্যায়। এই অধ্যায়টোৱে ছাত্ৰ-ছাত্ৰীসকলক বীজগণিতীয় ৰাশি (Algebraic Expressions) সহজ আৰু সৰল ৰূপত প্ৰকাশ কৰিবলৈ শিকায়। অনুশীলনী-14(B) ত উৎপাদক বিশ্লেষণৰ বিভিন্ন ধাৰণা আৰু পদ্ধতিৰ ওপৰত গুৰুত্ব আৰোপ কৰা হৈছে। এই অনুশীলনীটোৱে ছাত্ৰ-ছাত্ৰীসকলক গণিতৰ মৌলিক ধাৰণাসমূহ বুজিবলৈ আৰু সমস্যা সমাধান কৰাত সহায় কৰে।
1.তলত দিয়া ৰাশিবিলাক সূত্ৰ প্ৰয়োগ কৰি উৎপাদকত প্ৰকাশ কৰা –
(a) \(x^2 + 12x + 36\)
Solution:-
\(\Rightarrow x^2 + 12x + 36\)
\(\Rightarrow x^2 + 2 × x ×6+ 6^2\)
\(\Rightarrow (x+6)^2\)
\(\Rightarrow (x+6)(x+6)\)
(b) \(16 + 8x + x^2\)
Solutioon:-
\(\Rightarrow 16 + 8x + x^2\)
\(\Rightarrow 4^2 + 2 ×4×x+ x^2\)
\(\Rightarrow (4+x)^2\)
\(\Rightarrow (4+x)(4+x)\)
(c) \(4a^2 + 4ab + b^2\)
Solution:-
\(\Rightarrow 4a^2 + 4ab + b^2\)
\(\Rightarrow (2a)^2 + 2 × 2a ×b+ b^2\)
\(\Rightarrow (2a+b)^2\)
\(\Rightarrow (2a+b)(2a+b)\)
(d) \(25y^2 + 30y + 9\)
Solution:-
\(\Rightarrow 25y^2 + 30y + 9\)
\(\Rightarrow (5y)^2 + 2 × 5y × 3 + 3^2\)
\(\Rightarrow (5y+3)^2\)
\(\Rightarrow (5y+3)(5y+3)\)
(e) \(16u^2 + 24uv + 9v^2\)
Solution:-
\(\Rightarrow 16u^2 + 24uv + 9v^2\)
\(\Rightarrow (4u)^2 + 2 × 4u × 3v + (3v)^2\)
\(\Rightarrow (4u+3v)^2\)
\(\Rightarrow (4u+3v)(4u+3v)\)
(f) \(7p^2 + 4pq^2 – 70pq + 18p^2\)
Solution:-
\(\Rightarrow 7p^2 + 49q^2 – 70pq + 18p^2\)
\(\Rightarrow 25p^2 -70pq + 49q^2\)
\(\Rightarrow (5p)^2 – 2 × 5p ×7q+ 7q^2\)
\(\Rightarrow (5p-7q)^2\)
\(\Rightarrow (5p – 7q)(5p – 7q)\)
(g) \(a^4b^2 + 18a^2b – 81\)
Solution:-
\(\Rightarrow a^4b^2 + 18a^2b – 81\)
\(\Rightarrow (a^2b)^2 + 2 × a^2b × 9 + 9^2\)
\(\Rightarrow (a^2b + 9)^2\)
\(\Rightarrow (a^2b + 9)(a^2b + 9)\)
(l) \(4x^2 – 4x + y^2 – 6y + 10 – 2(2x-1)(y-3)\)
Solution:-
\(\Rightarrow 4x^2 – 4x + y^2 – 6y + 10 – 2(2x-1)(y-3)\)
\(\Rightarrow 4x^2 – 4x +1+ y^2 – 6y + 9 – 2(2x-1)(y-3)\)
\(\Rightarrow (2x)^2 – 2×2x×1 + 1^2 + y^2 – 2 × y × 3 + 3^2 – 2(2x-1)(y-3)\)
\(\Rightarrow(2x-1)^2+(y-3)^2 – 2(2x-1)(y-3)\)
\(\Rightarrow\{(2x-1)-(y-3)\}^2\)
\(\Rightarrow\{(2x – y + 2)\}^2\)
2. সূত্ৰৰ সহায়ত উৎপাদকত প্ৰকাশ কৰা-
(i) \(3p^2 – 27q^2\)
Solution:-
\(\Rightarrow 3(p^2 – 9q^2)\)
\(\Rightarrow 3(p^2 – (3q)^2)\)
\(\Rightarrow 3(p – 3q)(p – 3q)\)
(ii) \(4 – 4x + x^2 – Y^2\)
Solution:-
\(\Rightarrow 2^2 + 2×2×x + x^2 – y^2\)
\(\Rightarrow 2^2 + 2×2×x + x^2 – y^2\)
\(\Rightarrow (2-x)^2 – y^2\)
\(\Rightarrow (2-x + y)(2-x-y)\)
(iii) \(x^4 – 625\)
Solution:-
\(\Rightarrow ((x)^2)^2 – 25^2\)
\(\Rightarrow (x^2 + 25)(x^2-25)\)
\(\Rightarrow (x^2 + 25)(x^2-5^2)\)
\(\Rightarrow (x^2 + 25)((x-5)(x+5)\)
(iv) \(\frac{4x^2}{9}-\frac{4xy}{9}+ y^2 – 9\)
Solution:-
\(\Rightarrow \frac{4x^2}{9}-\frac{4xy}{9}+ y^2 – 9\)
\(\Rightarrow (\frac{2x}{3})^2-2×\frac{2x}{3}×y+ y^2 – 9\)
\(\Rightarrow (\frac{2x}{3} -y)^2 – 3^2\)
\(\Rightarrow (\frac{2x}{3} -y+3) × (\frac{2x}{3} -y+3)\)
(v) \((a+b)^2 – (a-b)^2\)
Solution:- \((a+b)^2 – (a-b)^2\)
\(\Rightarrow (a+b+a-b) (a+b-a+b)\)
\(\Rightarrow (a+b+a-b) (a+b-a+b)\)
\(\Rightarrow 2a× 2b\) = 4ab
(vi)\(25x^2 – 49y^2\)
Solution:- \(25x^2 – 49y^2\)
\(\Rightarrow (5x)^2 – (7y)^2\)
\(\Rightarrow (5x+7y) (5x-7y)\)
(vii) \(4a^2+9b^2-4c^2+12ab\)
Solution:- \(4a^2+9b^2-4c^2+12ab\)
\(\Rightarrow 4a^2+9b^2-4c^2+12ab\)
\(\Rightarrow 4a^2+12ab+9b^2-4c^2\)
\(\Rightarrow (2a)^2+ 2×2a×3b+(3b)^2-(2c)^2\)
\(\Rightarrow (2a-3b)^2-(2c)^2\)
\(\Rightarrow (2a-3b+2c)(2a-3b-2c)\)
(viii) \(a^6 – b^6\)
Solution:-
\(\Rightarrow a^6 – b^6\)
\(\Rightarrow (a^3)^2 – (b^3)^2\)
\(\Rightarrow (a^3+b^3)(a^3 + b^3)\)
\(\Rightarrow (a+b)(a^2- ab +b^2)(a-b)(a^2+ ab +b^2)\)
(ix) \(a^8 – b^8\)
Solution:-
\(\Rightarrow a^8 – b^8\)
\(\Rightarrow (a^4)^2 – (b^4)^2\)
\(\Rightarrow (a^4 + b^4)(a^4-b^4)\)
\(\Rightarrow (a^4 + b^4)(a^2+b^2)(a^2-b^2)\)
\(\Rightarrow (a^4 + b^4)(a^2+b^2)(a-b)(a+b)\)
(x) \(81 – x^2y^2\)
Solution:-
\(\Rightarrow (9^2-(xy)^2)\)
\(\Rightarrow (9-xy)(9+xy)\)
3. উৎপাদক বিশ্লেষণ কৰা —-
(i) \(3(a+b)+m(a+b)\)
Solution:-
\(\Rightarrow (a+b)(3+m)\)
(ii) \(px+x+qy+y\)
Solution:-
\(\Rightarrow x(p+1)+y(P+1)\)
\(\Rightarrow (p+1)(x+y)\)
(iii) \(8ac+4ad+6bc+3bd\)
Solution:-
\(\Rightarrow 8ac+6bc+4ad+3bd\)
\(\Rightarrow 2c(4a+3b)+d(4a+3b)\)
\(\Rightarrow (4a+3b)(2c+d)\)
(iv) \(2x^2-x + 2xy – y\)
Solution:-
\(\Rightarrow 2x^2 – x + 2xy – y\)
\(\Rightarrow x(2x – 1) + y(2x – 1)\)
\(\Rightarrow (2x – 1) (x+y)\)
(v) \(py^2 + p + (1+p^2)y\)
Solution:-
\(\Rightarrow py^2 + p + (1+p^2)y\)
\(\Rightarrow py^2 + p + y + p^2y\)
\(\Rightarrow py^2 + p^2y + p +y\)
\(\Rightarrow py(y + p) + 1(p +y)\)
\(\Rightarrow (py + 1) (p + y)\)
(vi) \(a^2x^3 + b^2y^2 + b^2x^3 + a^2Y^2\)
Solution:- \(a^2x^3 + b^2y^2 + b^2x^3 + a^2Y^2\)
\(\Rightarrow a^2x^3 + b^2y^2 + b^2x^3 + a^2Y^2\)
\(\Rightarrow a^2(x^3 + y^2) + b^2(y^2 + x^3) \)
\(\Rightarrow (x^3 + y^2) (a^2 + b^2) \)
(vii) lx(1+p) + my + nz + mpy + npz
Solution:-
\(\Rightarrow lx(1+p) + my + nz + mpy + npz\)
\(\Rightarrow lx(1+p) + my + mpy + npz + nz\)
\(\Rightarrow lx(1+p) + my (1+ p) + nz(1 +p)\)
\(\Rightarrow (1+p) (lx+my+nz)\)
(viii) \(a^2x + ax +2a + 2a^2 + x + 2\)
Solution:-
\(\Rightarrow a^2x + ax +2a + 2a^2 + x + 2\)
\(\Rightarrow a^2x + ax +x + 2a + 2a^2 + 2\)
\(\Rightarrow x(a^2 + a + 1) + 2(a + a^2 + 1)\)
\(\Rightarrow (a^2 + a + 1) (x+2)\)
(ix) \(8a^3 + 36a^2b + 54ab^2 + 27b^3\)
Solution:-
\(\Rightarrow 8a^3 + 36a^2b + 54ab^2 + 27b^3\)
\(\Rightarrow (2a)^3 + 3×(2a)^2×3b +3×(2a)×(3b)^2+ (3b)^3\)
\(\Rightarrow (2a+3b)^3\)
4. উৎপাদকত প্ৰকাশ কৰা —-
(i) \(x^2 + 9x +20\)
Solution:-
\(\Rightarrow x^2 + 9x + 20\)
\(\Rightarrow x^2 + 4x +5x + 20\)
\(\Rightarrow x( x+ 4) + 5(x + 4)\)
\(\Rightarrow ( x+ 4) (x + 5)\)
(ii) \(x^2 + 8x – 20\)
Solution:-
\(\Rightarrow x^2 + 8x – 20\)
\(\Rightarrow x^2 + 10x – 2x – 20\)
\(\Rightarrow x( x+ 10) – 2(x + 10)\)
\(\Rightarrow ( x+ 10) (x – 2)\)
(iii) \(y^2 – 5y – 36\)
Solution:-
\(\Rightarrow y^2 – 5y – 36\)
\(\Rightarrow y^2 – 9y + 4y – 36\)
\(\Rightarrow y( y – 9) + 4(y – 9)\)
\(\Rightarrow ( y – 9) (4 + y)\)
(iv) \(a^2 – 4a – 12\)
Solution:-
\(\Rightarrow a^2 – 4x – 12\)
\(\Rightarrow a^2 – 6a + 2a – 12\)
\(\Rightarrow a(a – 6) + 2(a – 6)\)
\(\Rightarrow (a – 6) (a + 2)\)
(v) \(u^2 – 21u +104\)
Solution:-
\(\Rightarrow u^2 – 21u + 104\)
\(\Rightarrow u^2 – 13u – 8u + 104\)
\(\Rightarrow u( u- 13) – 8(u – 13)\)
\(\Rightarrow ( u – 13) (u – 8)\)
(vi) \(x^2 + 2x – 3\)
Solution:-
\(\Rightarrow x^2 + 2x – 3\)
\(\Rightarrow x^2 + 3x – x – 3\)
\(\Rightarrow x(x+ 3) – 1(x + 3)\)
\(\Rightarrow ( x+ 3) (x – 1)\)
(vii) \(6x^2 + 5x + 1\)
Solution:-
\(\Rightarrow 6x^2 + 5x + 1\)
\(\Rightarrow 6x^2 + 2x + 3x + 1\)
\(\Rightarrow 2x(3x+ 1) + 1(x + 1)\)
\(\Rightarrow (3x+1) (2x + 1)\)
(viii)\(4x^2 + 25x – 21\)
Solution:-
\(\Rightarrow 4x^2 + 25x – 21\)
\(\Rightarrow 4x^2 + 28x – 3x – 21\)
\(\Rightarrow 4x(x+ 7) – 3(x + 7)\)
\(\Rightarrow ( x+ 7) (4x – 3)\)
(ix) \(12x^2 + 5x – 2\)
Solution:-
\(\Rightarrow 12x^2 + 5x – 2\)
\(\Rightarrow 12x^2 + 8x – 3x – 2\)
\(\Rightarrow 4x(3x+ 2) – 1 (3x + 2)\)
\(\Rightarrow (3x+2) (4x – 1)\)
(x) \(1 – x – 6x^2\)
Solution:-
\(\Rightarrow 1 – 3x + 2x – 6x^2\)
\(\Rightarrow 1(1-3x) + 2x (1 -3x)\)
\(\Rightarrow (1-3x)(1+ 2x)\)
(xi) \(18a^2 + 7ab – b^2\)
Solution:-
\(\Rightarrow 18a^2 + 7ab – b^2\)
\(\Rightarrow 18a^2 + 9ab – 2ab 3x – 2\)
\(\Rightarrow 9a(a+ b) – b (2a + b)\)
\(\Rightarrow (2a+b) (9- b)\)
(xii) \(2a^4 – a^2 – 1\)
Solution:-
\(\Rightarrow 2a^4 – a^2 – 1\)
\(\Rightarrow 2a^4 – 2a^2 + a^2 – 2\)
\(\Rightarrow 2a^2(a^2- 1) + 1 (a^2 – 1)\)
\(\Rightarrow (a^2-1) (2a^2 + 1)\)