নবম শ্রেণী গণিত সমাধান | 8. উৎপাদকে বিশ্লেষণ | কষে দেখি 8.3 সমাধান
নবম শ্রেণী কষে দেখি 8.3 সমাধান
1. উৎপাদকে বিশ্লেষণ করো: \(t^9-512\)
সমাধানঃ
\(t^9-512\)
\(=\left(t^3\right)^3-8^3\)
\(=(t^3-8)\{\left(t^3\right)^2+t^3\times8+\left(8\right)^2\}\)
\(=\{\left(t\right)^3-\left(2\right)^3\}(t^6+8t^3+64)\ \)
\(=(t-2)(t^2+2t+4)(\ t^6+8t^3+64)\)
2. উৎপাদকে বিশ্লেষণ করো: \(729p^6-\ q^6\)
সমাধানঃ
\(729p^6-\ q^6\)
\(=\left(27p^3\right)^2-\ \left(q^3\right)^2\)
\(=(27p^3+q^3)(27p^3-q^3)\)
\(={\left(3p\right)^3+q^3)}{\ \left(3p\right)^3-q^3)}\)
\(=\left(3p+q\right)\left(9p^2-18pq+q^2\right)\)
\(\times\left(3p-q\right)(9p^2+18pq+q^2)\)
\(=\left(3p+q\right)\left(3p-q\right)\)
\(\times\left(9p^2-18pq+q^2\right)(9p^2+18pq+q^2)\ \)
3. উৎপাদকে বিশ্লেষণ করো: \(8\left(p-3\right)^3+\ 343\)
সমাধানঃ
\(8\left(p-q\right)^3+\ 343\)
\(=\left\{2\left(p-3\right)\right\}^3+\ 7^3\)
\(=\left(2p-6\right)^3+7^3\)
\(=(2p-6+7)\{\left(2p-6\right)^2-(2p-6)\times7+7^2\}\)
\(=(2p+1)\{\left(2p\right)^2-2.2p.6+\left(6\right)^2-14p+42+49\}\)
\(=(2p+1)(4p^2-24p+36-14p+91)\ \)
\(=(2p+1)(4p^2-38p+127)\ \)
4. উৎপাদকে বিশ্লেষণ করো: \(\frac{1}{8a^3}+\frac{8}{b^3}\)
সমাধানঃ
\(\frac{1}{8a^3}+\frac{8}{b^3}\)
\(=\left(\frac{1}{2a}\right)^3+\left(\frac{2}{b}\right)^3\)
\(=\left(\frac{1}{2a}+\frac{2}{b}\right)\left\{\left(\frac{1}{2a}\right)^2-\frac{1}{2a}.\frac{2}{b}+\left(\frac{2}{b}\right)^2\right\}\)
\(=\left(\frac{1}{2a}+\frac{2}{b}\right)\left(\frac{1}{4a^2}-\frac{1}{ab}+\frac{4}{b^2}\right)\)
5. উৎপাদকে বিশ্লেষণ করো: \(\left(2a^3-b^3\right)^3-b^9\)
সমাধানঃ
\(\left(2a^3-b^3\right)^3-b^9\)
\(=\left(2a^3-b^3\right)^3-\left(b^3\right)^3\)
\(=(2a^3-b^3-b^3)\{\left(2a^3-b^3\right)^2\)
\(+(2a^3-b^3).b^3+\left(b^3\right)^2\}\)
\(=(2a^3-2b^3)\{(2a^3)^2-2.2a^3.b^3+(b^3)^2\)
\(+2a^3b^3-b^6+b^6\}\)
\(=2(a^3-b^3)(4a^6-4a^3b^3+b^6+2a^3b^3)\)
\(=2(a-b)(a^2+ab+b^2)(4a^6-2a^3b^3+b^6)\ \)
6. উৎপাদকে বিশ্লেষণ করো: \(AR^3-Ar^3+AR^2h-Ar^2h\)
সমাধানঃ
\(AR^3-Ar^3+AR^2h-Ar^2h\)
\(=A(R^3-r^3)+Ah(R^2-r^2)\ \)
\(=A\{(R^3-r^3)+h(R^2-r^2)\}\)
\(=A\{(R-r)(R^2+Rr+r^2)+h(R+r)(R-r)\}\)
\(=A(R-r)\{(R^2+Rr+r^2)+h(R+r)\}\)
\(=A(R-r)(R^2+Rr+r^2+Rh+rh)\)
7. উৎপাদকে বিশ্লেষণ করো: \(a^3+3a^2b+3ab^2+b^3-8\)
সমাধানঃ
\(a^3+3a^2b+3ab^2+b^3-8\)
\(=\left(a+b\right)^3-2^3\)
\(=(a+b-2)\{\left(a+b\right)^2+(a+b).2+2^2\}\)
\(=(a+b-2)\ (a^2+2ab+b^2+2a+2b+4)\)
8. উৎপাদকে বিশ্লেষণ করো: \(32x^4-500x\)
সমাধানঃ
\(32x^4-500x\)
\(=4x(8x^3-125)\)
\(=4x\{\left(2x\right)^3-\left(5\right)^3\}\)
\(=4x(2x-5)\{\left(2x\right)^2+2x.5+5^2\}\)
\(=4x(2x-5)(4x^2+10x+25)\)
9. উৎপাদকে বিশ্লেষণ করো: \(8a^3-b^3-4ax+2bx\)
সমাধানঃ
\(8a^3-b^3-4ax+2bx\)
\(=\left(2a\right)^3-b^3-2x(2a+b)\)
\(=(2a-b)\{\left(2a\right)^2+2a.b+b^2\}-2x(2a+b)\)
\(=(2a-b)(4a^2+2ab+b^2-2x)\)
10. উৎপাদকে বিশ্লেষণ করো: \(x^3-6x^2+12x-35\)
সমাধানঃ
\(x^3-6x^2+12x-35\)
\(=x^3-6x^2+12x-8-27\)
\(=x^3-3.x^2.2+3.x.2^2-23-3^3\)
\(=\left(x-2\right)^3-3^3\)
\(=(x-2-3)\{\left(x-2\right)^2+(x-2).3+3^2\}\)
\(=(x-5)(x^2-4x+4+3x-6+9)\)
\(=(x-5)(x^2-x+7)\)
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