mathematics, resolution into factors

1. Resolve into factor:

$\dfrac{1}{8}x^2-2$ 

Solution: 

    $\dfrac{1}{8}x^2-2$

$=\dfrac{x^2-16}{8}$

$=\dfrac{x^2-4^2}{8}$

$=\dfrac{(x+4)(x-4)}{8}$

$=\dfrac{1}{8}(x+4)(x-4)$ $(Ans.)$

Factorise similarly : $(i).\; \dfrac{1}{3}x^2-3$

                                             $(ii).\; \dfrac{1}{\sqrt{3}} x^3-9\sqrt{3}$            

2.Resolve into factor: 

$\dfrac{1}{2}a^3-4$

Solution:

    $\dfrac{1}{2}a^3-4$

$=\dfrac{a^3-8}{2}$

$=\dfrac{a^3-2^3}{2}$

$=\dfrac{(a-2)(a^2+a\cdot 2 +2^2)}{2}$

$=\dfrac{(a-2)(a^2+2a +4)}{2}$

$=\dfrac{1}{2} (a-2)(a^2+2a +4)$ $(Ans.)$

similar factorization: $\dfrac{1}{4}x^3-16$

3.Factorise the expression:

$x^2+\dfrac{1}{x^2}-x+\dfrac{1}{x}-2$

Solution:

    $x^2+\dfrac{1}{x^2}-x+\dfrac{1}{x}-2$

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

$=\left(x-\dfrac{1}{x}\right)^2+2\cdot x\cdot \dfrac{1}{x}-x+\dfrac{1}{x}-2$

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

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

$=\left(x-\dfrac{1}{x}\right)^2-\left(x-\dfrac{1}{x}\right)$

$=\left(x-\dfrac{1}{x}\right)\left(x-\dfrac{1}{x}-1\right)$

similarly factorise the expression:

$4y^2+\dfrac{1}{4y^2}+4y-\dfrac{1}{y}-2.$

4.Question:

$x^5+x^4+4x^3+4x^2+4x+4$

Solution:

  $x^5+x^4+4x^3+4x^2+4x+4$

$=x^4(x+1)+4x^2(x+1)+4(x+1)$

$=(x+1)\left(x^4+4x^2+4\right)$

$=(x+1)(x^2+1)^2$

$=(x+1)(x^2+1)(x^2+1)$          $(Ans)$

Similarly factorise:

$x^6-x^5+x^4-x^3+x^2-x$

Question-5:

$9x^2+\dfrac{1}{4x^2}-3-6x+\dfrac{1}{x}$

Solution:

   $9x^2+\dfrac{1}{4x^2}-3-6x+\dfrac{1}{x}$

$=9x^2-3+\dfrac{1}{4x^2}-6x+\dfrac{1}{x}$

$=(3x)^2-2\cdot 3x\cdot\dfrac{1}{2x}+\left(\dfrac{1}{2x}\right)^2-6x+\dfrac{1}{x}$

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

$=\left(3x-\dfrac{1}{2x}\right)\left(3x-\dfrac{1}{2x}-2\right)$    $(Ans.)$

Similarly:

$4x^2+\dfrac{9}{4x^2}+4x+\dfrac{1}{x}+6$

6.Factorize

$2x^2+\dfrac{1}{x^2}-3$

Solution:

   $2x^2+\dfrac{1}{x^2}-3$

$=2x^2-3+\dfrac{1}{x^2}$

$=2x^2-2-1+\dfrac{1}{x^2}$

$=2x\left(x-\dfrac{1}{x}\right)-\dfrac{1}{x}\left(x-\dfrac{1}{x}\right)$

$=\left(x-\dfrac{1}{x}\right)\left(2x-\dfrac{1}{x}\right)$        $(Ans)$

similarly:

$x^2-\dfrac{6}{x^2} +5$

9. Factorise:

$x^4-3x^2+9$

Solution:

   $x^4-3x^2+9$

$=\left(x^2\right)^2+2\cdot x^2\cdot 3+3^2-9x^2$

$=\left(x^2+3\right)^2-(3x)^2$

$=\left(x^2+3+3x\right)\left(x^2+3-3x\right)$

$=\left(x^2+3x+3\right)\left(x^2-3x+3\right)$     $(Ans)$

alternating Solution:

 $x^4-3x^2+9$

$=\left(x^2\right)^2+3^2-3x^2$

$=\left(x^2+3\right)^2-2\cdot x^2\cdot 3-3x^2$

$=\left(x^2+3\right)^2-6x^2-3x^2$

$=\left(x^2+3\right)^2-9x^2$

$=\left(x^2+3\right)^2-(3x)^2$

$=\left(x^2+3+3x\right)\left(x^2+3-3x\right)$

$=\left(x^2+3x+3\right)\left(x^2-3x+3\right)$

Similarly factorise the expressions:

$(i)$ $x^4-6x^2+25$

$(ii)$ $x^4-2x^2+49$

8. Factorise the expression:

$\dfrac{x^2}{a}-\dfrac{1}{a}+x^2-1$

Solution:

  $\dfrac{x^2}{a}-\dfrac{1}{a}+x^2-1$

$=\dfrac{x}{a}\left(x-\dfrac{\tfrac{1}{a}}{\tfrac{x}{a}}\right)+x\left(x-\dfrac{1}{x}\right)$

$=\dfrac{x}{a}\left(x-\dfrac{1}{x}\right)+x\left(x-\dfrac{1}{x}\right)$

$=\left(x-\dfrac{1}{x}\right)\left(\dfrac{x}{a}+x\right)$

$=x\left(x-\dfrac{1}{x}\right)\left(\dfrac{1}{a}+1\right)$

9. Question:

$a^2 x+ax^2+\dfrac{x}{a}-\dfrac{1}{ax}-a+1$

Solution:

  $a^2 x+ax^2+\dfrac{x}{a}-\dfrac{1}{ax}-a+1$

$= ax^2+\dfrac{x}{a}+a^2x+1-a-\dfrac{1}{ax}$

$=x^2\left(a+\dfrac{1}{ax}\right)+ax\left(a+\dfrac{1}{ax}\right)-1\left(a+\dfrac{1}{ax}\right)$

$=\left(a+\dfrac{1}{ax}\right)\left(x^2+ax-1\right)$   $(Ans.)$

Question-10:

$x^4-13x^2+36$

Solution:

     $x^4-13x^2+36$

$=\left(x^2\right)^2+6^2-13x^2$

$=\left(x^2+6\right)^2-2\cdot x^2\cdot 6-13x^2$

$=\left(x^2+6\right)^2-12x^2-13x^2$

$=\left(x^2+6\right)^2-25x^2$

$=\left(x^2+6\right)^2-(5x)^2$

$=\left(x^2+6+5x\right)\left(x^2+6-5x\right)$

$=\left(x^2+5x+6\right)\left(x^2-5x+6\right)$

$=\left(x^2+3x+2x+6\right)\left(x^2-3x-2x+6\right)$

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

$=(x+2)(x-2)(x+3)(x-3)$        (Ans.)

Similar Question:

$x^4-9x^2+64$

Question-11:

$x^4+x^2+25$

Question-12:

$x^2+\left(\dfrac{a+b}{a b}\right) x+\dfrac{1}{a b}$

Solution:

$x^2+\left(\dfrac{a+b}{a b}\right) x+\dfrac{1}{a b}$

$=x^2+\left(\dfrac{1}{a}+\dfrac{1}{b}\right) x+\dfrac{1}{a b}$

$=x^2+\dfrac{1}{a} x+\dfrac{1}{b} x+\dfrac{1}{a b} $

$=x\left(x+\dfrac{1}{a}\right)+\dfrac{1}{b}\left(x+\dfrac{1}{a}\right) $

$=\left(x+\dfrac{1}{a}\right)\left(x+\dfrac{1}{b}\right) $

Factorise the expressions:

13. $99 a^{2}-202 a b+99 b^{2}$

14. $2(2 x+y)^{2}-5(2 x+y)+3$

15. $a x^{2}+(a b-1) x-b$

16. $21 x^{2}+40 x y-21 y^{2}$

17. $12 x^{2}+65 x+77$

18. $8 x-3-4 x^{2}$

19.$17 x-7 x^{2}-6$

20.$12 x^{2}+7 x-12$

21.$14 x-3 x^{2}+5$

22. $3(a+b)^{2}-2 a-2 b-8$

23. $2 a^{6}-13 a^{3}-24$

24. $8 a^{4}+2 a^{2}-45$

25. $3(x+y)^{2}-10(x+y)(x-y)+3(x-y)^{2}$

26. $12 x^{3}-7 x^{2}-10 x$

27. $6 x^{2} y+x y^{2}-15 y^{3}$

28. $6(a+b)^{2}+5\left(a^{2}-b^{2}\right)-6(a-b)^{2}$

29. $30 x^{2} y-33 x y-18 y$.

30. $2 x^{2}-3 x-(a+1)(2 a-1)$

31. $a^{2} x^{2}+a x-(a+2)(a+1)$

32. $(x+y) t^{2}+2 x t+x-y$

33. $\left(a^{2}-a-2\right) x^{2}-3 x-1$

34. $a(a+1) x^{2}-x-a(a-1)$

35. $\left(a^{2}+1\right) x^{2}+a^{2}\left(a^{2}+2\right) x-\left(a^{2}+1\right)$

36. $2 a^{2}+b^{2}-c^{2}+3 a b+a c$

37. $2\left(a^{2}+\frac{1}{a^{2}}\right)-\left(a-\frac{1}{a}\right)-7$

38. $2 a^{2}+a-3 a b-b+b^{2}$

39. $x^2-3 a x+\dfrac{5}{4} a^2$