গণিত প্রকাশ উৎপাদকে বিশ্লেষণ (Class-9) কষে দেখি 8.West Bengal Board Class 9 Math Solution Chapter 8 || Class 9 Solution koshe dekhi 8.2 || উৎপাদকে বিশ্লেষণ || Ganit Prakash Class 9 Solution koshe dekhi 8.2 || Ganit Prakash Solution Class 9 In Bengali || গণিত প্রকাশ সমাধান নবম শ্রেণী || 2 || West Bengal Board Class 9 Math Solution Chapter 8 || উৎপাদকে বিশ্লেষণ || WBBSE Class 9 Math koshe dekhi 8.2
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Class 9 Solution koshe dekhi 8.2 || Class 9 Chapter 8 || Ganit Prakash Class 9 Solution || West Bengal Board Class 9 Math || Class 9 Chapter 8 koshe dekhi 8.2 || নবম শ্রেণী কষে দেখি 8.2 || উৎপাদকে বিশ্লেষণ
কষে দেখি - 8.2
Class 9 Solution koshe dekhi 8.2 || Class 9 Chapter 8 || Ganit Prakash Class 9 Solution || West Bengal Board Class 9 Math || Class 9 Chapter 8 koshe dekhi 8.2 || নবম শ্রেণী কষে দেখি 8.2 || উৎপাদকে বিশ্লেষণ
নীচের বীজগাণিতিক সংখ্যামালাগুলিকে উৎপাদকে বীশ্লেশণ করি :
1. \(\frac{x^{4}}{16}-\frac{y^{4}}{81}\)
\( \frac{x^{4}}{16}-\frac{y^{4}}{81}\)
\(=\left(\frac{x^{2}}{4}\right)^{2}-\left(\frac{y^{2}}{9}\right)^{2} \)
\(=\left(\frac{x^{2}}{4}+\frac{y^{2}}{9}\right)\left(\frac{x^{2}}{4}-\frac{y^{2}}{9}\right)\)
\(=\left(\frac{x^{2}}{4}+\frac{y^{2}}{9}\right)\left\{\left(\frac{x}{2}\right)^{2}-\left(\frac{y}{3}\right)^{2}\right\}\)
\(=\left(\frac{x^{2}}{4}+\frac{y^{2}}{9}\right)\left(\frac{x}{2}+\frac{y}{3}\right)\left(\frac{x}{2}-\frac{y}{3}\right) \)
\(=\left(\frac{x^{2}}{4}\right)^{2}-\left(\frac{y^{2}}{9}\right)^{2} \)
\(=\left(\frac{x^{2}}{4}+\frac{y^{2}}{9}\right)\left(\frac{x^{2}}{4}-\frac{y^{2}}{9}\right)\)
\(=\left(\frac{x^{2}}{4}+\frac{y^{2}}{9}\right)\left\{\left(\frac{x}{2}\right)^{2}-\left(\frac{y}{3}\right)^{2}\right\}\)
\(=\left(\frac{x^{2}}{4}+\frac{y^{2}}{9}\right)\left(\frac{x}{2}+\frac{y}{3}\right)\left(\frac{x}{2}-\frac{y}{3}\right) \)
2. \(m^{2}+\frac{1}{m^{2}}+2-2 m-\frac{2}{m}\)
\(m^{2}+\frac{1}{m^{2}}+2-2 m-\frac{2}{m} \)
\(=(m)^{2}+\left(\frac{1}{m}\right)^{2}+2 \cdot m \cdot \frac{1}{m}-2\left(m+\frac{1}{m}\right) \)
\(=\left(m+\frac{1}{m}\right)^{2}-2\left(m+\frac{1}{m}\right)\)
\(=\left(m+\frac{1}{m}\right)\left(m+\frac{1}{m}-2\right) \)
\(=\left(m+\frac{1}{m}\right)\left\{\frac{m^{2}+1-2 m}{m}\right\}\)
\(=\frac{1}{m}\left(m+\frac{1}{m}\right)(m-1)^{2} \)
\(=\frac{1}{m^{2}}\left(m^{2}+1\right)(m-1)(m-1)\)
\(=(m)^{2}+\left(\frac{1}{m}\right)^{2}+2 \cdot m \cdot \frac{1}{m}-2\left(m+\frac{1}{m}\right) \)
\(=\left(m+\frac{1}{m}\right)^{2}-2\left(m+\frac{1}{m}\right)\)
\(=\left(m+\frac{1}{m}\right)\left(m+\frac{1}{m}-2\right) \)
\(=\left(m+\frac{1}{m}\right)\left\{\frac{m^{2}+1-2 m}{m}\right\}\)
\(=\frac{1}{m}\left(m+\frac{1}{m}\right)(m-1)^{2} \)
\(=\frac{1}{m^{2}}\left(m^{2}+1\right)(m-1)(m-1)\)
3. \(9 p^{2}-24 p q+16 q^{2}+3 a p-4 a q\)
\(9 p^{2}-24 p q+16 q^{2}+3 a p-4 a q \)
\(=(3 p)^{2}-2.3 p \cdot 4 q+(4 q)^{2}+a(3 p-4 q) \)
\(=(3 p-4 q)^{2}+a(3 p-4 q)\)
\(=(3 p-4 q)(3 p-4 q+a)\)
\(=(3 p)^{2}-2.3 p \cdot 4 q+(4 q)^{2}+a(3 p-4 q) \)
\(=(3 p-4 q)^{2}+a(3 p-4 q)\)
\(=(3 p-4 q)(3 p-4 q+a)\)
4. \(4 x^{4}+81\)
\(4 x^{4}+81\)
\(=\left(2 x^{2}\right)^{2}+(9)^{2} \)
\(=\left(2 x^{2}+9\right)^{2}-2.2 x^{2} .9\)
\(=\left(2 x^{2}+9\right)^{2}-36 x^{2}\)
\(=\left(2 x^{2}+9\right)^{2}-(6 x)^{2}\)
\(=\left(2 x^{2}+9+6 x\right)\left(2 x^{2}+9-6 x\right) \)
\(=\left(2 x^{2}+6 x+9\right)\left(2 x^{2}-6 x+9\right) \)
\(=\left(2 x^{2}\right)^{2}+(9)^{2} \)
\(=\left(2 x^{2}+9\right)^{2}-2.2 x^{2} .9\)
\(=\left(2 x^{2}+9\right)^{2}-36 x^{2}\)
\(=\left(2 x^{2}+9\right)^{2}-(6 x)^{2}\)
\(=\left(2 x^{2}+9+6 x\right)\left(2 x^{2}+9-6 x\right) \)
\(=\left(2 x^{2}+6 x+9\right)\left(2 x^{2}-6 x+9\right) \)
5. \( x^{4}-7 x^{2}+1\)
\( x^{4}-7 x^{2}+1 \)
\(=\left(x^{2}\right)^{2}+2 \cdot x^{2} \cdot 1+(1)^{2}-9 x^{2} \)
\(=\left(x^{2}+1\right)^{2}-(3 x)^{2}\)
\(=\left(x^{2}+1+3 x\right)\left(x^{2}+1-3 x\right)\)
\(=\left(x^{2}+3 x+1\right)\left(x^{2}-3 x+1\right) \)
\(=\left(x^{2}\right)^{2}+2 \cdot x^{2} \cdot 1+(1)^{2}-9 x^{2} \)
\(=\left(x^{2}+1\right)^{2}-(3 x)^{2}\)
\(=\left(x^{2}+1+3 x\right)\left(x^{2}+1-3 x\right)\)
\(=\left(x^{2}+3 x+1\right)\left(x^{2}-3 x+1\right) \)
6. \(p^{4}-11 p^{2} q^{2}+q^{4}\)
\( p^{4}-11 p^{2} q^{2}+q^{4} \)
\(=\left(p^{2}\right)^{2}-2 \cdot p^{2} \cdot q^{2}+\left(q^{2}\right)^{2}-9 p^{2} q^{2}\)
\(=\left(p^{2}-q^{2}\right)^{2}-(3 p q)^{2}\)
\(=\left(p^{2}-q^{2}+3 p q\right)\left(p^{2}-q^{2}-3 p q\right) \)
\(=\left(p^{2}+3 p q-q^{2}\right)\left(p^{2}-3 p q-q^{2}\right) \)
\(=\left(p^{2}\right)^{2}-2 \cdot p^{2} \cdot q^{2}+\left(q^{2}\right)^{2}-9 p^{2} q^{2}\)
\(=\left(p^{2}-q^{2}\right)^{2}-(3 p q)^{2}\)
\(=\left(p^{2}-q^{2}+3 p q\right)\left(p^{2}-q^{2}-3 p q\right) \)
\(=\left(p^{2}+3 p q-q^{2}\right)\left(p^{2}-3 p q-q^{2}\right) \)
7. \(a^{2}+b^{2}-c^{2}-2 a b\)
\( a^{2}+b^{2}-c^{2}-2 a b\)
\(=a^{2}-2 a b+b^{2}-c^{2} \)
\(=(a-b)^{2}-(c)^{2}\)
\(=(a-b+c)(a-b-c) \)
\(=a^{2}-2 a b+b^{2}-c^{2} \)
\(=(a-b)^{2}-(c)^{2}\)
\(=(a-b+c)(a-b-c) \)
8. \(3 a(3 a+2 c)-4 b(b+c)\)
\( 3 a(3 a+2 c)-4 b(b+c) \)
\(=9 a^{2}+6 a c-4 b^{2}-4 b c \)
\(=9 a^{2}+6 a c-4 b^{2}-4 b c+c^{2}-c^{2} \)
\( =(3 a)^{2}+2 \cdot 3 a \cdot c+c^{2}-\left(4 b^{2}+4 b c+c^{2}\right) \)
\(=(3 a+c)^{2}-\left\{(2 b)^{2}+2 \cdot 2 b \cdot c+(c)^{2}\right\} \)
\(=(3 a+c)^{2}-(2 b+c)^{2} \)
\(=(3 a+c+2 b+c)(3 a+c-2 b-c) \)
\(=(3 a+2 b+2 c)(3 a-2 b)\)
\(=9 a^{2}+6 a c-4 b^{2}-4 b c \)
\(=9 a^{2}+6 a c-4 b^{2}-4 b c+c^{2}-c^{2} \)
\( =(3 a)^{2}+2 \cdot 3 a \cdot c+c^{2}-\left(4 b^{2}+4 b c+c^{2}\right) \)
\(=(3 a+c)^{2}-\left\{(2 b)^{2}+2 \cdot 2 b \cdot c+(c)^{2}\right\} \)
\(=(3 a+c)^{2}-(2 b+c)^{2} \)
\(=(3 a+c+2 b+c)(3 a+c-2 b-c) \)
\(=(3 a+2 b+2 c)(3 a-2 b)\)
9. \(a^{2}-6 a b+12 b c-4 c^{2}\)
\( a^{2}-6 a b+12 b c-4 c^{2} \)
\(=a^{2}-6 a b+9 b^{2}-9 b^{2}+12 b c-4 c^{2} \)
\(=(a)^{2}-2 \cdot a \cdot 3 b+(3 b)^{2}-\left\{(3 b)^{2}-2 \cdot 3 b \cdot 2 c+(2 c)^{2}\right\}\)
\(=(a-3 b)^{2}-(3 b-2 c)^{2} \)
\(=(a-3 b+3 b-2 c)(a-3 b-3 b+2 c)\)
\(=(a-2 c)(a-6 b+2 c) \)
\(=a^{2}-6 a b+9 b^{2}-9 b^{2}+12 b c-4 c^{2} \)
\(=(a)^{2}-2 \cdot a \cdot 3 b+(3 b)^{2}-\left\{(3 b)^{2}-2 \cdot 3 b \cdot 2 c+(2 c)^{2}\right\}\)
\(=(a-3 b)^{2}-(3 b-2 c)^{2} \)
\(=(a-3 b+3 b-2 c)(a-3 b-3 b+2 c)\)
\(=(a-2 c)(a-6 b+2 c) \)
10. \(3 a^{2}+4 a b+b^{2}-2 a c-c^{2}\)
\( 3 a^{2}+4 a b+b^{2}-2 a c-c^{2} \)
\(=4 a^{2}+4 a b+b^{2}-a^{2}-2 a c-c^{2} \)
\(=(2 a)^{2}+2 \cdot 2 a \cdot b+(b)^{2}-\left\{(a)^{2}+2 \cdot a \cdot c+(c)^{2}\right\} \)
\(=(2 a+b)^{2}-(a+c)^{2} \)
\(=(2 a+b+a+c)(2 a+b-a-c) \)
\(=(3 a+b+c)(a+b-c) \)
\(=4 a^{2}+4 a b+b^{2}-a^{2}-2 a c-c^{2} \)
\(=(2 a)^{2}+2 \cdot 2 a \cdot b+(b)^{2}-\left\{(a)^{2}+2 \cdot a \cdot c+(c)^{2}\right\} \)
\(=(2 a+b)^{2}-(a+c)^{2} \)
\(=(2 a+b+a+c)(2 a+b-a-c) \)
\(=(3 a+b+c)(a+b-c) \)
11. \(x^{2}-y^{2}-6 a x+2 a y+8 a^{2}\)
\( x^{2}-y^{2}-6 a x+2 a y+8 a^{2} \)
\(=x^{2}-6 a x+9 a^{2}-y^{2}+2 a y-a^{2} \)
\(=(x)^{2}-2 \cdot x \cdot 3 a+(3 a)^{2}-\left(y^{2}-2 a y+a^{2}\right)\)
\(=(x-3 a)^{2}-(y-a)^{2} \)
\(=(x-3 a+y-a)(x-3 a-y+a) \)
\(=(x+y-4 a)(x-y-2 a) \)
\(=x^{2}-6 a x+9 a^{2}-y^{2}+2 a y-a^{2} \)
\(=(x)^{2}-2 \cdot x \cdot 3 a+(3 a)^{2}-\left(y^{2}-2 a y+a^{2}\right)\)
\(=(x-3 a)^{2}-(y-a)^{2} \)
\(=(x-3 a+y-a)(x-3 a-y+a) \)
\(=(x+y-4 a)(x-y-2 a) \)
12. \(a^{2}-9 b^{2}+4 c^{2}-25 d^{2}-4 a c+30 b d\)
\(a^{2}-9 b^{2}+4 c^{2}-25 d^{2}-4 a c+30 b d\)
\(=a^{2}-4 a c+4 c^{2}-9 b^{2}+30 b d-25 d^{2} \)
\(=(a)^{2}-2 \cdot a \cdot 2 c+(2 c)^{2}-\left\{(3 b)^{2}-2 \cdot 3 b \cdot 5 d+(5 d)^{2}\right\}\)
\(=(a-2 c)^{2}-(3 b-5 d)^{2} \)
\(=(a-2 c+3 b-5 d)(a-2 c-3 b+5 d) \)
\(=(a+3 b-2 c-5 d)(a-3 b-2 c+5 d)\)
\(=a^{2}-4 a c+4 c^{2}-9 b^{2}+30 b d-25 d^{2} \)
\(=(a)^{2}-2 \cdot a \cdot 2 c+(2 c)^{2}-\left\{(3 b)^{2}-2 \cdot 3 b \cdot 5 d+(5 d)^{2}\right\}\)
\(=(a-2 c)^{2}-(3 b-5 d)^{2} \)
\(=(a-2 c+3 b-5 d)(a-2 c-3 b+5 d) \)
\(=(a+3 b-2 c-5 d)(a-3 b-2 c+5 d)\)
13. \(3 \mathrm{a}^{2}-\mathrm{b}^{2}-\mathrm{c}^{2}+2 \mathrm{ab}-2 \mathrm{bc}+2 \mathrm{ca}\)
\(3 a^{2}-b^{2}-c^{2}+2 a b-2 b c+2 a c \)
\(=4 a^{2}-\left[a^{2}+b^{2}+c^{2}-2 a b+2 b c-2 c a\right]\)
\(=4 a^{2}-\left[(b)^{2}+(c)^{2}+(-a)^{2}+2 \cdot b \cdot c+2 \cdot c \cdot(-a)+2 \cdot(-a) \cdot b]\right.\)
\( =(2 a)^{2}-(b+c-a)^{2} \)
\(=(2 a+b+c-a)(2 a-b-c+a) \)
\(=(a+b+c)(3 a-b-c) \)
\(=4 a^{2}-\left[a^{2}+b^{2}+c^{2}-2 a b+2 b c-2 c a\right]\)
\(=4 a^{2}-\left[(b)^{2}+(c)^{2}+(-a)^{2}+2 \cdot b \cdot c+2 \cdot c \cdot(-a)+2 \cdot(-a) \cdot b]\right.\)
\( =(2 a)^{2}-(b+c-a)^{2} \)
\(=(2 a+b+c-a)(2 a-b-c+a) \)
\(=(a+b+c)(3 a-b-c) \)
14. \(x^{2}-2 x-22499\)
\(x^{2}-2 x-22499 \)
\(=x^{2}-2 \cdot x \cdot 1+(1)^{2}-22499-1 \)
\(=(x-1)^{2}-22500\)
\(=(x-1)^{2}-(150)^{2} \)
\(=(x-1+150)(x-1-150)\)
\(=(x+149)(x-151) \)
\(=x^{2}-2 \cdot x \cdot 1+(1)^{2}-22499-1 \)
\(=(x-1)^{2}-22500\)
\(=(x-1)^{2}-(150)^{2} \)
\(=(x-1+150)(x-1-150)\)
\(=(x+149)(x-151) \)
15. \(x^{2}-y^{2}\left(a^{2}-b^{2}\right)+4 a b x y\)
\(\left(x^{2}-y^{2}\right)\left(a^{2}-b^{2}\right)+4 a b x y \)
\(=x^{2} a^{2}-x^{2} b^{2}-y^{2} a^{2}+y^{2} b^{2}+2 a b x y+2 a b x y\)
\(=(x a)^{2}+2 \cdot x a \cdot y b+(y b)^{2}-\left[(x b)^{2}-2 \cdot x b \cdot a y+(a y)^{2}\right] \)
\(=(x a+y b)^{2}-(x b-a y)^{2} \)
\(=(x a+y b+x b-a y)(x a+y b-x b+a y)\)
\(=x^{2} a^{2}-x^{2} b^{2}-y^{2} a^{2}+y^{2} b^{2}+2 a b x y+2 a b x y\)
\(=(x a)^{2}+2 \cdot x a \cdot y b+(y b)^{2}-\left[(x b)^{2}-2 \cdot x b \cdot a y+(a y)^{2}\right] \)
\(=(x a+y b)^{2}-(x b-a y)^{2} \)
\(=(x a+y b+x b-a y)(x a+y b-x b+a y)\)
Class 9 Solution koshe dekhi 8.2 || Class 9 Chapter 8 || Ganit Prakash Class 9 Solution || West Bengal Board Class 9 Math || Class 9 Chapter 8 koshe dekhi 8.2 || নবম শ্রেণী কষে দেখি 8.2 || উৎপাদকে বিশ্লেষণ
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