নবম শ্রেণী গণিত সমাধান | 8. উৎপাদকে বিশ্লেষণ | কষে দেখি 8.4 সমাধান
নবম শ্রেণী কষে দেখি 8.4 সমাধান
1. উৎপাদকে বিশ্লেষণ করো:
\(x^3+y^3-12xy+64\)
সমাধানঃ
\(x^3+y^3-12xy+64\)
\(=x^3+y^3+4^3-3.x.y.4\)
\(=(x+y+4)(x^2+y^2+4^2-xy-4y-4x)\)
\(=(x+y+4)(x^2+y^2+16-xy-4y-4x)\)
2. উৎপাদকে বিশ্লেষণ করো:
\(8x^3-y^3+1+6xy\)
সমাধানঃ
\(8x^3-y^3+1+6xy\)
\(=\ \left(2x\right)^3+\left(-y\right)^3+\left(1\right)^3-3\times2x\times(-y)\times1\)
\(=\ (2x-y+1)\{\left(2x\right)^2+\left(-y\right)^2+\left(1\right)^2\)
\(-2x.(-y)-(-y).1-1.2x\}\)
\(=\ (2x-y+1)(4x^2+y^2+1+2xy+y-2x)\)
3. উৎপাদকে বিশ্লেষণ করো:
\(8a^3-27b^3-1-18ab\)
সমাধানঃ
\(8a^3-27b^3-1-18ab\)
\(=\left(2a\right)^3+\left(-3b\right)^3+\left(-1\right)^3\)
\(+3\times2a\times(-3b)\times(-1)\)
\(=(2a-3b-1)\{\left(2a\right)^2+\left(-3b\right)^2+\left(-1\right)^2\)
\(-2a.(-3b)-(-3b)(-1)-2a.(-1)\}\)
\(=(2a-3b-1)(4a^2+9b^2+1+6ab-3b+2a)\)
4. উৎপাদকে বিশ্লেষণ করো:
\(1+8x^3+18xy-27y^3\)
সমাধানঃ
\(1+8x^3+18xy-27y^3\)
\(=\left(1\right)^3+\left(2x\right)^3+\left(-3y\right)^3-3.1.2x.(-3y)\)
\(=(1+2x-3y)\{\left(1\right)^2+\left(2x\right)^2+\left(-3y\right)^2-1.2x\)
\(-2x.(-3y)-1.(-3y)\}\)
\(=(1+2x-3y)(1+4x^2+9y^2-2x+6xy+3y)\)
5. উৎপাদকে বিশ্লেষণ করো:
\(\left(3a-2b\right)^3+\left(2b-5c\right)^3+\left(5c-3a\right)^3\)
সমাধানঃ
ধরি, 3a-2b=p, 2b-5c=q এবং 5c-3a=r
p+q+r=3a-2b+2b-5c+5c-3a=0
\(\left(3a-2b\right)^3+\left(2b-5c\right)^3+\left(5c-3a\right)^3\)
\(=p^3+q^3+r^3\)
=3pqr [যেহেতু, p+q+r=0,
সুতরাং \(p^3+q^3+r^3=3pqr\)]
=3(3a-2b)(2b-5c)(5c-3a)
6. উৎপাদকে বিশ্লেষণ করো:
\(\left(2x-y\right)^3-\left(x+y\right)^3+\left(2y-x\right)^3\)
সমাধানঃ
ধরি, 2x-y=p, -(x+y)=q এবং 2y-x=r
p+q+r=2x-y-(x+y)+2y-x=0
\(\left(2x-y\right)^3-\left(x+y\right)^3+\left(2y-x\right)^3\)
\(=\left(2x-y\right)^3+{-{\left(x+y\right)}}^3+\left(2y-x\right)^3\)
\(=p^3+q^3+r^3\)
=3pqr [যেহেতু, p+q+r=0,
সুতরাং \(p^3+q^3+r^3=3pqr\)]
=3(2x-y)(x+y)(2y-x)
7. উৎপাদকে বিশ্লেষণ করো:
\(a^6+32a^3-64\)
সমাধানঃ
\(a^6+32a^3-64\)
\(=a^6+8a^3-64\ +\ 24a^3\)
\(=\ \left(a^2\right)^3+\left(2a\right)^3+\left(-4\right)^3-3.a^2.2a.(-4)\)
\(=(a^2+2a-4)\{\left(a^2\right)^2+\left(2a\right)^2+\left(-4\right)^2-a^2.2a\)
\(-2a.(-4)-(-4).a^2\}\)
\(=\ (a^2+2a-4)(a^4+4a^2+16-\ 2a^3+8a+4a^2)\)
\(=\ (a^2+2a-4)(a^4-2a^3+8a^2+8a+16)\)
8. উৎপাদকে বিশ্লেষণ করো:
\(a^6-18a^3+125\)
সমাধানঃ
\(a^6-18a^3+125\)
\(=a^6+27a^3+125-45a^3\)
\(=\left(a^2\right)^3+\left(3a\right)^3+\left(5\right)^3-3.a^2.3a.5\)
\(=(a^2+3a+5)\{\left(a^2\right)^2+\left(3a\right)^2+\left(5\right)^2\)
\(-a^2.3a-3a.5-5.a^2\}\)
\(=(a^2+3a+5)(a^4+9a^2+25-3a^3-15a-5a^2)\)
\(=(a^2+3a+5)(a^4-3a^3+4a^2-15a-25)\)
9. উৎপাদকে বিশ্লেষণ করো:
\(p^3\left(q-r\right)^3+q^3\left(r-p\right)^3+r^3\left(p-q\right)^3\)
সমাধানঃ
ধরি, p(q-r)=a, q(r-p)=b এবং r(p-q)=c
a+b+c=p(q-r)+q(r-p)+r(p-q)
=pq-qr-qr-pq+pr-qr=0
\(\ p^3\left(q-r\right)^3+q^3\left(r-p\right)^3+r^3\left(p-q\right)^3\)
\(=\left\{p\left(q-r\right)\right\}^3+\left\{q\left(r-p\right)\right\}^3+\left\{r\left(p-q\right)\right\}^3\)
\(=a^3+b^3+c^3\)
=3abc [যেহেতু, a+b+c=0,
সুতরাং \(a^3+b^3+c^3=3abc\)]
=3p(q-r)q(r-p)r(p-q)
=3pqr(p-q)(q-r)(r-p)
10. উৎপাদকে বিশ্লেষণ করো:
\(p^3+\frac{1}{p^3}+\frac{\mathrm{26}\ }{\mathrm{27}}\)
সমাধানঃ
\({\ \ \ \ p}^3+\frac{1}{p^3}+\frac{26}{27}\)
\(=p^3+\frac{1}{p^3}+\frac{27-1}{27}\)
\(=p^3+\frac{1}{p^3}+1-\frac{1}{\mathrm{27}}\)
\(=p^3+\frac{1}{p^3}-\frac{1}{\mathrm{27}}+1\)
\(=p^3+\left(\frac{1}{p}\right)^3+\left(-\frac{1}{3}\right)^3-3\times\ p\times\frac{1}{p}\times\left(-\frac{1}{3}\right)\)
\(=\left(p+\frac{1}{p}-\frac{1}{3}\right)\)
\(\times\left\{p^2+\left(\frac{1}{p}\right)^2+\left(-\frac{1}{3}\right)^2-p.\frac{1}{p}-\left(\frac{1}{p}\right).\left(-\frac{1}{3}\right)-\left(-\frac{1}{3}\right).p\right\}\)
\(=\left(p+\frac{1}{p}-\frac{1}{3}\right)\left(p^2+\frac{1}{p^2}+\frac{1}{9}-1+\frac{1}{\mathrm{3p}}+\frac{p}{3}\right)\)
\(=\left(p+\frac{1}{p}-\frac{1}{3}\right)\left(p^2+\frac{1}{p^2}-\frac{8}{9}+\frac{1}{\mathrm{3p}}+\frac{p}{3}\right)\)
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