নবম শ্রেণী গণিত সমাধান | 8. উৎপাদকে বিশ্লেষণ | কষে দেখি 8.1 সমাধান
নীচের বীজগাণিতিক সংখ্যামালাকে উৎপাদকে বিশ্লেষণ করিঃ
1. \(\frac{x^4}{16}-\frac{y^4}{81}\)
সমাধানঃ
\(\frac{x^4}{16}-\frac{y^4}{81}\)
=\(\left(\frac{x^2}{4}\right)^2-\left(\frac{y^2}{9}\right)^2\)
=\(\left(\frac{x^2}{4}+\frac{y^2}{9}\right)\left(\frac{x^2}{4}-\frac{y^2}{9}\right)\)
=\(\left(\frac{x^2}{4}+\frac{y^2}{9}\right)\left\{\left(\frac{x}{2}\right)^2-\left(\frac{y}{3}\right)^2\right\}\)
=\(\left(\frac{x^2}{4}+\frac{y^2}{9}\right)\left(\frac{x}{2}+\frac{y}{2}\right)\left(\frac{x}{2}-\frac{y}{2}\right)\)
2. \(m^2+\frac{1}{m^2}+2-2m-\frac{2}{m}\)
সমাধানঃ
\(m^2+\frac{1}{m^2}+2-2m-\frac{2}{m}\)
\(=\left(m\right)^2+\left(\frac{1}{m}\right)^2-2\left(m+\frac{1}{m}\right)+2\)
\(=\left(m+\frac{1}{m}\right)^2-2.m\mathrm{.}\frac{1}{m}-2\left(m+\frac{1}{m}\right)+2\)
\(=\left(m+\frac{1}{m}\right)^2-2-2\left(m+\frac{1}{m}\right)+2\)
\(=\left(m+\frac{1}{m}\right)^2-2\left(m+\frac{1}{m}\right)\)
\(=\left(m+\frac{1}{m}\right)\left(m+\frac{1}{m}-2\right)\)
3. \(9p^2-24pq+16q^2+3ap-4aq\)
সমাধানঃ
\(9p^2-24pq+16q^2+3ap-4aq\)
\(=\left(3p\right)^2-2.3p.4q+\left(4q\right)^2+a(3p-4q)\)
\(=\left(3p-4q\right)^2\ +a(3p-4q)\)
\(=(3p-4q)(3p-4q+a)\)
4. \(4x^4+81\)
সমাধানঃ
\(4x^4+81\)
\(=\left(2x^2\right)^2+\left(9\right)^2\)
\(=\left(2x^2+9\right)^2-2.2x^2.9\)
\(=\left(2x^2+9\right)^2-36x^2\)
\(=\left(2x^2+9\right)^2-\left(6x\right)^2\)
\(=(2x^2+9+6x)\ (2x^2+9-6x)\)
\(=(2x^2+6x+9)\ (2x^2-6x+9)\ \)
5. \(x^4-7x^2+1\)
সমাধানঃ
\(x^4-7x^2+1\)
\(=x^4+2x^2+1-9x^2\)
\(=\left(x^2\right)^2+2.x^2.1+\left(1\right)^2-\left(3x\right)^2\)
\(=\left(x^2+1\right)^2-\left(3x\right)^2\)
\(=(x^2+1+3x)\ (x^2+1-3x)\)
\(=(x^2+3x+1)\ (x^2\ -3x+1)\)
6. \(p^4-11p^2q^2-q^4\)
সমাধানঃ
\(p^4-11p^2q^2-q^4\)
\(=p^4-2p^2q^2-q^4-9p^2q^2\)
\(=\left(p^2\right)^2-2p^2q^2-\left(q^2\right)^2-9p^2q^2\)
\(=\left(p^2-q^2\right)^2-\left(3pq\right)^2\)
\(=(p^2-q^2+3pq)(p^2-q^2-3pq)\)
\(=\left(p^2+3pq-q^2\right)(p^2-3pq-q^2)\)
7. \(a^2+b^2-c^2-2ab\)
সমাধানঃ
\(a^2+b^2-c^2-2ab\)
\(=a^2-2ab+b^2-c^2\)
\(=\left(a-b\right)^2-c^2\)
\(=(a-b+c)(a-b-c)\)
8. 3a(3a+2c)-4b(b+c)
সমাধানঃ
\(3a(3a+2c)-4b(b+c)\ \)
\(=9a^2+6ac-4b^2-4bc\)
\(=\left(3a\right)^2-\left(2b\right)^2+6ac-4bc\)
\(=(3a+2b)(3a-2b)+2c(3a-2b)\)
\(=(3a-2b)(3a+2b+2c)\)
9. \(a^2-6ab+12bc-4c^2\)
সমাধানঃ
\(a^2-6ab+12bc-4c^2\)
\(=a^2-\left(2c\right)^2-6ab+12bc\)
\(=(a+2c)(a-2c)-6b(a-2c)\)
\(=(a-2c)(a+2c-6b)\)
\(=(a-2c)(a-6b+2c)\)
10. \(3a^2+4ab+b^2-2ac-c^2\)
সমাধানঃ
\(3a^2+4ab+b^2-2ac-c^2\)
\(=4a^2-a^2+4ab+b^2-2ac-c^2\)
\(=\left(2a\right)^2+2.2a.b+b^2-(a^2+2ac+c^2)\)
\(=\left(2a+b\right)^2-\left(a+c\right)^2\)
\(=\{(2a+b)+(a+c)\}\{(2a+b)-(a+c)\}\)
\(=(2a+b+a+c)(2a+b-a-c)\ \)
\(=(3a+b+c)(a+b-c)\)
11. \(x^2-y^2-6ax+2ay+8a^2\)
সমাধানঃ
\(x^2-y^2-6ax+2ay+8a^2\)
\(=x^2-6ax+9a^2-y^2+2ay-a^2\)
\(=x^2-2.x.3a+\left(3a\right)^2-(\ y^2-2ay+a^2)\)
\(=\left(x-3a\right)^2-\left(y-a\right)^2\)
\(=\{(x-3a)+(y-a)\}\{(x-3a)-(y-a)\}\)
\(=(x-3a+y-a)(x-3a-y+a)\)
\(=(x+y-4a)(x-y-2a)\)
12. \(a^2-9b^2+4c^2-25d^2-4ac+30bd\)
সমাধানঃ
\(a^2-9b^2+4c^2-25d^2-4ac+30bd\)
\(=a^2-4ac+4c^2-9b^2+30bd-25d^2\)
\(=a^2-2.a.2c+\left(2c\right)^2-{\left(3b\right)^2-2.3b.5d-+\left(5d\right)^2}\)
\(=\left(a-2c\right)^2-\left(3b-5d\right)^2\)
\(=\{(a-2c)+(3b-5d)\}\{(a-2c)-(3b-5d)\}\)
\(=(a-2c+3b-5d)(a-2c-3b+5d)\)
\(=(a+3b-2c-5d)(a-3b-2c+5d)\)
13. \(3a^2-b^2-c^2+2ab-2bc+2ca\)
সমাধানঃ
\(3a^2-b^2-c^2+2ab-2bc+2ca\)
\(=3a^2-b^2-c^2+3ab-ab-bc-bc+3ca-ca\)
\(=3a^2+3ab+3ca-ab-b^2-bc-ca-bc-c^2\)
\(=3a(a+b+c)-b(a+b+c)-c(a+b+c)\ \)
\(=(a+b+c)(3a-b-c)\)
14. \(x^2-2x-22499\)
সমাধানঃ
\(x^2-2x-22499\)
\(=x^2-2x+1-22500\)
\(=\left(x-1\right)^2-\left(150\right)^2\)
\(=(x-1+150)(x-1-150)\)
\(=(x+149)(x-151)\)
15. \((x^2-y^2)(a^2-b^2)+4abxy\)
সমাধানঃ
\((x^2-y^2)(a^2-b^2)+4abxy\)
\(=a^2x^2-a^2y^2-b^2x^2-b^2y^2+2abxy+2bxy\)
\(=\left(ax\right)^2+2.ax.by+\left(by\right)^2\)
\(-{\left(ay\right)^2+2.ay.bx+\left(bx\right)^2}\)
\(=\left(ax+by\right)^2-\left(ay-bx\right)^2\)
\(=\{(ax+by)+(ay-bx)\}\{(ax+by)-(ay-bx)\}\)
\(=(ax+by+ay-bx)(\ ax+by-ay+bx)\)
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