Class-VII-Factorization-WB: - LET US LEARN MATHEMATICS

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Class-VII

K. C. NAG-প্রশ্নমালা -4.4

উৎপাদক বিশ্লেষণ করোঃ

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1. $a^4+4$

সমাধানঃ

$\\a^4+4\\\\=(a^2)^2+2.a^2.2+(2)^2-4.a^2\\\\=(a^2+2)^2-(2a)^2\\\\=(a^2+2+2a)(a^2+2-2a)\\\\=(a^2+2a+2)(a^2-2a+2)$

2. $4x^4+81$

সমাধানঃ

$\\4x^4+81\\\\=(2x^2)^2+2.2x^2.9+(9)^2-36x^2\\\\=(2x^2+9)^2-(6x)^2\\\\=(2x^2+9+6x)(2x^2+9-6x)\\\\=(2x^2+6x+9)(2x^2-6x+9)$

3. $81x^4+64y^4$

সমাধানঃ

$\\81x^4+64y^4\\\\=(9x^2)^2+2.9x^2.8y^2+(8y^2)^2-144x^2y^2\\\\=(9x^2+8y^2)^2-(12xy)^2\\\\=(9x^2+8y^2+12xy)(9x^2+8y^2-12xy)\\\\=(9x^2+12xy+8y^2)(9x^2-12xy+8y^2)$

4. $9x^4+36$

সমাধানঃ

$\\9x^4+36\\\\=(3x^2)^2+2.3x^2.6+(6)^2-36x^2\\\\=(3x^2+6)^2-(6x)^2\\\\=(3x^2+6+6x)(3x^2+6-6x)\\\\=(3x^2+6x+6)(3x^2-6x+6)\\\\=3 \times 3 \times (x^2+2x+2)(x^2-2x+2)$

5. $x^4+x^2+1$

সমাধানঃ

$x^4+x^2+1\\\\= (x^2)^2+2.x^2.1+1-x^2\\\\=(x^2+1)^2-(x)^2\\\\=(x^2+x+1)(x^2-x+1)$

6. $a^4+a^2y^2+y^4$

সমাধানঃ

$a^4+a^2y^2+y^4\\\\=(a^2)^2+2.a^2.y^2+(y^2)^2-a^2.y^2\\\\=(a^2+y^2)^2-(ay)^2\\\\=(a^2+y^2-ay)(a^2+y^2+ay)\\\\=(a^2-ay+y^2)(a^2+ay+y^2)$

7. $x^8+x^4+1$

সমাধানঃ

$x^8+x^4+1\\\\=(x^4)^2+2.x^4.1+1-x^4\\\\=(x^4+1)^2-(x^2)^2\\\\=(x^4+1-x^2)(x^4+1+x^2)\\\\=(x^4-x^2+1)(x^4+x^2+1)\\\\=(x^4-x^2+1)(x^4+x^2+1)\\\\=(x^4-x^2+1)\left \{ (x^2)^2+2.x^2.1+1-x^2 \right \}\\\\=(x^4-x^2+1)\left \{ (x^2+1)^2-(x)^2 \right \}\\\\=(x^4-x^2+1)(x^2-x+1)(x^2+x+1)$

8. $x^4-6x^2+1$

সমাধানঃ

$x^4-6x^2+1\\\\=(x^2)^2-2.x^2.1+(1)^2-4x^2\\\\=(x^2-1)^2-(2x)^2\\\\=(x^2-1-2x)(x^2-1+2x)\\\\=(x^2-2x-1)(x^2+2x-1)$

9. $a^8+a^4x^4+x^8$

সমাধানঃ

$a^8+a^4x^4+x^8\\\\=(a^4)^2+2.a^4x^4+(x^4)^2-a^4x^4\\\\=(a^4+x^4)^2-(a^2x^2)^2\\\\=(a^4+x^4-a^2x^2)(a^4+x^4+a^2x^2)\\\\=(a^4-a^2x^2+x^4)\left \{ \left ( a^2 \right )^2+2.a^2.x^2+(x^2)^2-a^2x^2 \right \}\\\\=(a^4-a^2x^2+x^4)\left \{ \left ( a^2 +x^2\right )^2-(ax)^2\right \}\\\\=(a^4-a^2x^2+x^4)(a^2+ax+x^2)(a^2-ax+x^2)$

10. $p^4+2p^2+9$

সমাধানঃ

$p^4+2p^2+9\\\\=(p^2)^2+2.p^2.3+(3)^2-4p^2\\\\=(p^2+3)^2-(2p)^2\\\\=(p^2-2p+3)(p^2+2p+3)$

11. $x^4-23x^2y^2+y^4$

সমাধানঃ

$x^4-23x^2y^2+y^4\\\\=(x^2)^2+2.x^2.y^2+(y^2)^2-25x^2y^2\\\\=(x^2+y^2)^2-(5xy)^2\\\\=(x^2+y^2-5xy)(x^2+y^2+5xy)\\\\=(x^2-5xy+y^2)(x^2+5xy+y^2)$

12. $4x^4y^4-16x^2y^2+9$

সমাধানঃ

$4x^4y^4-16x^2y^2+9\\\\=(2x^2y^2)^2-2.2x^2y^2.3+(3)^2-4x^2y^2\\\\=(2x^2y^2-3)^2-(2xy)^2\\\\=(2x^2y^2-3-2xy)(2x^2y^2-3+2xy)\\\\=(2x^2y^2-2xy-3)(2x^2y^2+2xy-3)$

13. $4x^4+625$

সমাধানঃ

$4x^4+625\\\\=(2x^2)^2+(25)^2\\\\=(2x^2)^2+2.2x^2.25+(25)^2-100x^2\\\\=(2x^2+25)^2-(10x)^2\\\\=(2x^2+25-10x)(2x^2+25+10x)\\\\=(2x^2-10x+25)(2x^2+10x+25)$

14. $9y^4-33y^2+16$

সমাধানঃ

$9y^4-33y^2+16\\\\=(3y^2)^2-2.3y^2.4+(4)^2-9y^2\\\\=(3y^2+4)^2-(3y)^2\\\\=(3y^2+4-3y)(3y^2+4+3y)\\\\=(3y^2-3y+4)(3y^2+3y+4)$

15. $9a^4-a^2+16$

সমাধানঃ

$9a^4-a^2+16\\\\=(3a^2)^2+2.3a^2.4+(4)^2-24a^2-a^2\\\\=(3a^2+4)^2-25a^2\\\\=(3a^2+4)^2-(5a)^2\\\\=(3a^2-5a+4)(3a^2+5a+4)$

16. $4x^4+11x^2+25$

সমাধানঃ

$4x^4+11x^2+25\\\\=(2x^2)^2+2.2x^2.5+(5)^2-9x^2\\\\=(2x^2+5)^2-(3x)^2\\\\=(2x^2-3x+5)(2x^2+3x+5)$

17. $4x^4-24x^2+25$

সমাধানঃ

$4x^4-24x^2+25\\\\=(2x^2)^2-2.2x^2.5+(5)^2-4x^2\\\\=(2x^2-5)^2-(2x)^2\\\\=(2x^2-5-2x)(2x^2-5+2x)\\\\=(2x^2-2x-5)(2x^2+2x-5)$

18. $4x^4-16x^2+25$

সমাধানঃ

$4x^4-16x^2+25\\\\=4x^4+20x^2+25-36x^2\\\\=\left \{ (2x^2)^2+2.2x^2.5+(5)^2 \right \}-(6x)^2\\\\=(2x^2+5)^2-(6x)^2\\\\=(2x^2+5+6x)(2x^2+5-6x)\\\\=(2x^2+6x+5)(2x^2-6x+5)$

19. $4a^4+1$

সমাধানঃ

$4a^4+1\\\\=(2a^2)^2+2.2a^2.1+(1)^2-4a^2\\\\=(2a^2+1)^2-(2a)^2\\\\=(2a^2-2a+1)(2a^2+2a+1)$

20. $x^4+8x^2+144$

সমাধানঃ

$x^4+8x^2+144\\\\=(x^2)^2+2.x^2.12+(12)^2-16x^2\\\\=(x^2+12)^2-(4x)^2\\\\=(x^2+4x+12)(x^2-4x+12)$

21. $(a+b)^4+4$

সমাধানঃ

$(a+b)^4+4\\\\=\left \{ \left ( a+b \right )^{2} \right \}^{2}+2.\left ( a+b \right )^{2}.2+\left ( 2 \right )^{2}-4\left ( a+b \right )^{2}\\\\=\left \{ \left ( a+b \right )^{2} +2\right \}^{2}-\left ( 2a+2b \right )^{2}\\\\=\left \{ \left ( a+b \right )^{2} +2-\left ( 2a+2b \right )\right \}\left \{ \left ( a+b \right )^{2} +2+\left ( 2a+2b \right )\right \}\\\\=\left \{ \left ( a+b \right )^{2}-\left ( 2a+2b \right ) +2\right \}\left \{ \left ( a+b \right )^{2}+\left ( 2a+2b \right ) +2\right \}$

22. $(x^2-2xy)-(z^2-2yz)$

সমাধানঃ

$(x^2-2xy)-(z^2-2yz)\\\\=x^2-2xy-z^2+2yz\\\\=x^2-z^2-2xy+2yz\\\\=\left \{ \left ( x \right )^{2} -\left ( z \right )^{2}\right \}-2y\left ( x-z \right )\\\\=(x-z)(x+z)-2y(x-z)\\\\=(x-z)(x+z-2y)$

23. $4a^2-9b^2-c^2-6bc$

সমাধানঃ

$4a^2-9b^2-c^2-6bc\\\\= (2a)^2-(9b^2+6bc+c^2)\\\\=(2a)^2-(3b+c)^2\\\\=(2a-3b-c)(2a+3b+c)$

24. $36x^2+y^2-z^2-12xy$

সমাধানঃ

$36x^2+y^2-z^2-12xy\\\\=(6x)^2-2.6x.y+(y)^2-(z)^2\\\\=(6x-y)^2-(z)^2\\\\=(6x-y+z)(6x-y-z)$

25. $4p^2-1+9q^2-25r^2+12pq-10r$

সমাধানঃ

$4p^2-1+9q^2-25r^2+12pq-10r\\\\=(2p)^2+2.2p.3q+(3q)^2-1-(5r)^2-10r\\\\=(2p+3q)^2-\left \{ (5r)^2+10r+1\right \}\\\\=(2p+3q)^2-(5r+1)^2\\\\=(2p+3q+5r+1)(2p+3q-5r-1)$

26. $16x^2-16y^2-9z^2-24x+24yz+9$

সমাধানঃ

$16x^2-16y^2-9z^2-24x+24yz+9\\\\=\left \{ (4x)^2-2.4x.3+(3)^2 \right \}-( 16y^2-24yz+9z^2)\\\\=\left \{ (4x)^2-2.4x.3+(3)^2 \right \}-\left \{ (4y)^2-2.4y.3z+(3z)^2 \right \}\\\\=\left \{ (4x-3)^2 \right \}-\left \{ (4y-3z)^2 \right \}\\\\=(4x-3-4y+3z)(4x-3+4y-3z)$

27. $2(ab+cd)-a^2-b^2+c^2+d^2$

সমাধানঃ

$2(ab+cd)-a^2-b^2+c^2+d^2\\\\=2ab+2cd-a^2-b^2+d^2\\\\=\left ( c^2+2cd+d^2 \right )-\left ( a^2-2ab +b^2\right )\\\\=\left ( c+d \right )^{2}-\left ( a-b \right )^2\\\\=\left ( c+d+ a-b \right )\left ( c+d-a+b \right )$

28. $4x^2-3y^2-z^2-4xy+4yz$

সমাধানঃ

$4x^2-3y^2-z^2-4xy+4yz\\\\=(2x)^2-2.2x.y+(y)^2-\left ( z^2-4yz+4y^2 \right )\\\\=\left \{(2x)^2-2.2x.y+(y)^2 \right \}-\left \{ (z)^2-2.z.2y+(2y)^2 \right \}\\\\=\left ( 2x-y \right )^2-\left ( z-2y \right )^2\\\\=\left ( 2x-y-z+2y \right )\left ( 2x-y+z-2y \right )\\\\=\left ( 2x-z+y \right )\left ( 2x+z-3y \right )$

29. $4a^2-4ab-c^2-2bc$

সমাধানঃ

$4a^2-4ab-c^2-2bc\\\\=(2a)^2-2.2a.b+(b)^2-(b)^2-c^2-2bc\\\\=(2a-b)^2-(b^2+2bc+c^2)\\\\=(2a-b)^2-(b+c)^2\\\\=(2a-b+b+c)(2a-b-b-c)\\\\=(2a+c)(2a-2b-c)$

30. $3a^2+b^2-c^2-4ab+2ac$

সমাধানঃ

$3a^2+b^2-c^2-4ab+2ac\\\\=4a^2-a^2+b^2-c^2-4ab+2ac\\\\=\left ( 4a^2-4ab+b^2 \right )-\left ( a^2-2ac+c^2 \right )\\\\=\left \{ (2a)^2-2.2a.b+(b)^2 \right \}-\left \{ (a)^2-2.a.c+(c)^2 \right \}\\\\=(2a-b)^2-(a-c)^2\\\\=(2a-b+a-c)(2a-b-a+c)\\\\=(3a-b-c)(a-b+c)$

31. $a^2-b^2-6bc-9c^2$

সমাধানঃ

$a^2-b^2-6bc-9c^2\\\\=a^2-(b^2+6bc+9c^2)\\\\=(a)^2-(b+3c)^2\\\\=(a+b+3c)(a-b-3c)$

32. $a^2+2a-b^2+2b$

সমাধানঃ

$a^2+2a-b^2+2b\\\\=a^2+2a+1-1-b^2+2b\\\\=\left ( a^2+2a+1 \right )-\left (1-2b+b^2 \right )\\\\=\left ( a+1 \right )^2-\left ( 1-b \right )^2\\\\=\left ( a+1+1-b \right )\left ( a+1-1+b \right )\\\\=\left ( a-b+2 \right )\left ( a+b \right )$

33. $a^2-b^2-c^2+d^2+2(bc+ab)$

সমাধানঃ

$a^2-b^2-c^2+d^2+2(bc+ad)\\\\=\left ( a^2+2ad+d^2 \right )-\left ( b^2-2bc+c^2 \right )\\\\=\left ( a+d \right )^{2}-\left ( b-c \right )^{2}\\\\=\left ( a+d+b-c \right )\left ( a+d-b+c \right )\\\\=\left ( a+b-c+d \right )\left ( a-b+c+d \right )$

34. $x^2-y^2-6xa+2ya+8a^2$

সমাধানঃ

$x^2-y^2-6xa+2ya+8a^2\\\\=x^2-6xa+9a^2-y^2+2ya-a^2\\\\=\left ( x^2-6xa+9a^2 \right )-\left (y^2-2ya+a^2 \right )\\\\=(x-3a)^2-(y-a)^2\\\\=(x-3a-y+a)(x-3a+y-a)\\\\=(x-y-2a)(x+y-4a)$

35. $(x^2-1)(y^2-1)+4xy$

সমাধানঃ

$(x^2-1)(y^2-1)+4xy\\\\=x^2y^2-x^2-y^2+1+4xy\\\\=(xy)^2+2.xy.1+(1)^2-(x^2-2xy+y^2)\\\\=(xy+1)^2-(x-y)^2\\\\=(xy+1+x-y)(xy+1-x+y)$

36. $7a^4-88a^2+121$

সমাধানঃ

$7a^4-88a^2+121\\\\=16a^4-88a^2+121-9a^4\\\\=(4a^2)^2-2.4a^2.11+(11)^2-(3a^2)^2\\\\=(4a^2-11)^2-(3a^2)^2\\\\=(4a^2-11-3a^2)(4a^2-11+3a^2)\\\\=(a^2-11)(7a^2-11)$

37. $5a^4-4a^2b^2-b^4$

সমাধানঃ

$5a^4-4a^2b^2-b^4\\\\=9a^4-4a^4-4a^2b^2-b^4\\\\=(3a^2)^2-\left \{ (2a^2)^2+2.2a^2.b^2+(b^2)^2 \right \}\\\\=(3a^2)^2-(2a^2+b^2)^2\\\\=(3a^2-2a^2-b^2)(3a^2+2a^2+b^2)\\\\=(a^2-b^2)(5a^2+b^2)\\\\=(a+b)(a-b)(5a^2+b^2)$

38. $x^4+x^2-2$

সমাধানঃ

$x^4+x^2-2\\\\=x^4+x^3+2x^2+2x-x^3-x^2-2x-2\\\\=x(x^3+x^2+2x+2)-1(x^3+x^2+2x+2)\\\\=(x^3+x^2+2x+2)(x-1)\\\\=\left \{ x^2(x+1)+2(x+1) \right \}(x-1)\\\\=(x+1)(x^2+2)(x-1)=(x^2+2)(x+1)(x-1)$

39. $\frac{a}{b}+ \frac{b}{a}+2$

সমাধানঃ

$\frac{a}{b}+ \frac{b}{a}+2\\\\=\frac{a^2+b^2}{ab}+2\\\\=\frac{a^2+b^2+2ab}{ab}\\\\=\frac{(a+b)^2}{ab}=\frac{(a+b)(a+b)}{ab}\\\\=\left ( \frac{a+b}{a} \right )\left ( \frac{a+b}{b} \right )\\\\=\left ( 1+\frac{b}{a} \right )\left ( \frac{a}{b}+1 \right )$

40. $2a^2c^2+2c^2b^2+2a^2b^2-a^4-b^4-c^4$ সমাধানঃ

$(a^2-b^2-c^2)^2=a^4+b^4+c^4-2a^2b^2+2b^2c^2-2a^2c^2\\\\\Rightarrow a^4+b^4+c^4-2a^2b^2-2a^2c^2=(a^2-b^2-c^2)^2-2b^2c^2$

$2a^2c^2+2c^2b^2+2a^2b^2-a^4-b^4-c^4\\\\=-(a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2)\\\\=-(a^4+b^4+c^4-2a^2b^2-2c^2a^2-2b^2c^2)\\\\=-\left \{ (a^2-b^2-c^2)^2-2b^2c^2- 2b^2c^2\right \}\\\\=4b^2c^2-(a^2-b^2-c^2)^2\\\\=(2bc)^2-(a^2-b^2-c^2)^2\\\\=(2bc-a^2+b^2+c^2)(2bc+a^2-b^2-c^2)\\\\=\left \{ (b^2+2bc+c^2)-a^2 \right \}\left \{ a^2-(b^2-2bc+c^2) \right \}\\\\=\left \{ (b+c)^2-a^2 \right \}\left \{ a^2-(b-c)^2 \right \}\\\\=(a+b+c)(b+c-a)(a+b-c)(c+a-b)$

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Solution of sample questions -VII

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