Question 1: Factorize:

(i) \displaystyle  xyz-2xy = xy(z-2)  [HCF = xy]

(ii) \displaystyle  35x^3y-49xy^2 = 7xy(5x^2-7y)  [HCF = 7xy]

(iii) \displaystyle  p(2a-1)+q(1-2a)

\displaystyle  = p(2a-1)-q(2a-1)

\displaystyle  = (2a-1)(p-q) c  [HCF = (2a-1)]

(iv) \displaystyle  a(a+b)^3-3a^2b(a+b)

\displaystyle  = a(a+b) \Big[ (a+b)^2 - 3ab \Big]

\displaystyle  = a(a+b)(a^2+2ab+b^2-3ab)

\displaystyle  = a(a+b)(a^2+b^2-ab )  [HCF = a(a+b)]

(v) \displaystyle  10x(2a+b)^3-15y(2a+b)^2 + 35(2a+b)

\displaystyle  5(2a+b) \Big[ 2x(2a+b)^2 - 3y(2a+b) + 7 \Big]  [HCF = 5(2a+b)]

\displaystyle  \\

Question 2: Factorize

(i) \displaystyle  x^3 -2x^2y+3xy^2-6y^3

\displaystyle  = x^2(x-2y)+3y^2(x-2y)

\displaystyle  = (x-2y)(x^2+3y^2)

(ii) \displaystyle  a^2x^2+(ax^2+1)x+a

\displaystyle  = a^2x^2+ax^3+x+a

\displaystyle  = ax^2(a+x)+(a+x)

\displaystyle  = (a+x)(ax^2+1)

(iii) \displaystyle  x^2+y-xy-x

\displaystyle  = x(x-y) -(x-y)

\displaystyle  = (x-y)(x-1)

(iv) \displaystyle  ab(x^2+y^2)+xy(a^2+b^2)

\displaystyle  = abx^2+aby^2+xya^2+xyb^2

\displaystyle  = ax(bx+ay)+by(ay+bx)

\displaystyle  = (bx+ay)(ax+by)

(v) \displaystyle  a^3+ab(1-2a)-2b^2

\displaystyle  = a^2(a-2b)+b(a-2b)

\displaystyle  = (a-2b)(a^2+b)

\displaystyle  \\

Question 3: Factorize

(i) \displaystyle  \frac{a^2}{4b^2}  -  \frac{1}{3}  +  \frac{b^2}{9a^2}

\displaystyle  = (  \frac{a}{2b})^2  - 2 \times  \frac{a}{2b}  \times  \frac{b}{3a}  + (  \frac{b}{3a})^2

\displaystyle  = \Big(  \frac{a}{2b}  - \frac{b}{3a}  \Big)^2

(ii) \displaystyle  \Big( x-  \frac{1}{x}  \Big)^2 + 6 \Big( x-  \frac{1}{x}  \Big) + 9

\displaystyle  =  \Big( x-  \frac{1}{x}  \Big)^2 + 2 \times 3 \times \Big( x-  \frac{1}{x}  \Big) + (3)^3

\displaystyle  =  \Big( x-  \frac{1}{x}  + 3 \Big)^2

(iii) \displaystyle  \Big( x^2 +  \frac{1}{x^2}  \Big)^2 -4 \Big( x+  \frac{1}{x}  \Big) + 6

\displaystyle  =  \Big( x+  \frac{1}{x}  \Big)^2 -2 - 4 \Big( x+  \frac{1}{x}  \Big) + 6

\displaystyle  =  \Big( x+  \frac{1}{x}  \Big)^2 - 4 \Big( x+  \frac{1}{x}  \Big) + 4

\displaystyle  =  \Big( x +  \frac{1}{x}  -2 \Big)^2

(iv) \displaystyle  2a^2 + 2 \sqrt{6}ab + 3b^2

\displaystyle  = (\sqrt{2}a)^2 + 2 \sqrt{2} \sqrt{3} ab + (\sqrt{3}b)^2

\displaystyle  = (\sqrt{2}a + \sqrt{3}b)^2

(v) \displaystyle  4(x-y)^2-12(x-y)(x+y)+9(x+y)^2

\displaystyle  = [2(x-y)]^2 - 2 . 2(x-y) . [3(x+y)]^2 + [3(x+y)]^2

\displaystyle  = \Big( 2(x-y) - 3(x+y) \Big)

\displaystyle  = (-x-5y)^2 = (x+5y)^2

\displaystyle  \\

Question 4: Factorize: 

Note: We are going to use the following formulas in this set of exercise: \displaystyle  {(a+b)}^2=\ a^2+2ab+b^2 and \displaystyle  {(a-b)}^2=\ a^2-2ab+b^2

(i) \displaystyle  p^2q^2-p^4q^4

\displaystyle  = p^2q^2(1-p^2q^2)

\displaystyle  = p^2q^2(1-pq)(1+pq)

(ii) \displaystyle  3x^4 - 243

\displaystyle  = 3(x^4-81)

\displaystyle  = 3(x^2-9)(x^2+9)

\displaystyle  = 3(x-3)(x+3)(x^2+9)

(iii) \displaystyle  25x^2-10x+1-36y^2

\displaystyle  = (5x-1)^2 - (6y)^2

\displaystyle  = (5x-1-6y)(5x-1+6y)

(iv) \displaystyle  x^4 - 625

\displaystyle  = (x^2-25)(x^2+25)

\displaystyle  = (x-5)(x+5)(x^2+25)

(v) \displaystyle  16(2x-1)^2 - 25y^2

\displaystyle  = \Big[ 4(2x-1) \Big]^2 - (5y)^2

\displaystyle  = \Big[ 4(2x-1) - 5y \Big] \Big[ 4(2x-1) + 5y \Big]

\displaystyle  = (8x-4-5y)(8x-4+5y)

(vi) \displaystyle  x^4-(2y-3z)^2

\displaystyle  = (x^2)^2 - (2y-3z)^2

\displaystyle  = (x^2-2y+3z)(x^2+2y-3z)

(vii) \displaystyle  x^4-14x^2+1

\displaystyle  = (x^2+1)^2 - 2x^2 - 14x^2

\displaystyle  = (x^2+1)^2 - 16x^2

\displaystyle  = (x^2+1-4x)(x^2+1+4x)

(viii) \displaystyle  x^2-y^2-4xz +4z^2

\displaystyle  = x^2 - 4zx+4z^2 - y^2

\displaystyle  = (x-2z)^2 - y^2

\displaystyle  = (x-2z-y)(x-2z+y)

(ix) \displaystyle  a^2(b+c)-(b+c)^3

\displaystyle  = (b+c) \Big[ a^2 - (b+c)^2 \Big]

\displaystyle  = (b+c) (a-b-c)(a+b+c)

(x) \displaystyle  x^4+y^4-7x^2y^2

\displaystyle  = x^4 +y^4 - 2x^2y^2 - 9x^2y^2

\displaystyle  = (x^2-y^2)^2 - (3xy)^2

\displaystyle  = (x^2-y^2 - 3xy)(x^2-y^2+3xy)

\displaystyle  \\

Question 5: Factorize:

Note: We are going to use the following formulas in this set of exercise: \displaystyle  a^3+b^3 = (a+b)(a^2 -ab + b^2) and \displaystyle  a^3-b^3 = (a-b)(a^2 +ab + b^2)

(i) \displaystyle  27a^3+8

\displaystyle  = (3a)^3+(2)^3

\displaystyle  = (3a+2)(9a^2-6a+4)

(ii) \displaystyle  1-27a^3

\displaystyle  = (1)^3-(3a)^3

\displaystyle  = (1-3a)(1+3a+9a^2)

(iii) \displaystyle  a-b -a^3+b^3

\displaystyle  = (a-b) - (a^3-b^3)

\displaystyle  = (a-b) -(a-b)(a^2+ab+b^2)

\displaystyle  = (a-b)(1-a^2-ab-b^2)

(iv) \displaystyle  (2x+3y)^3 - (2x-3y)^3

\displaystyle  = (2x+3y-2x+3y) \Big[ (2x+3y)^2 + (2x+3y)(2x-3y) + (2x-3y)^2 \Big]

\displaystyle  = 6y(4x^2+12xy+9y^2 + 4x^2 - 9y^2 + 4x^2-12xy+9y^2 )

\displaystyle  = 6y(12x^2+9y^2) = 18y(4x^2+3y^2)

(v) \displaystyle  (a+b)^3 - 8(a-b)^3

\displaystyle  = (a+b)^3 - 2^3(a-b)^3

\displaystyle  =(a+b-2a+2b) \big[ (a+b)^2 + 2(a+b)(a-b) + 4(a-b)^2 \Big]

\displaystyle  = (3b-a)(a^2 + 2ab + b^2 + 2a^2 - 2b^2 + 4a^2 -8ab + 4b^2 )

\displaystyle  = (3b-a)(7a^2-6ab+3b^2)

(vi) \displaystyle  a^{12} + b^{12}

\displaystyle  = (a^4)^3+(b^4)^3

\displaystyle  = (a^4+b^4)(a^8-a^4b^4+b^8)

(vii) \displaystyle  x^3-12x(x-4)-64

\displaystyle  = x^3 - 12x(x-4)-4^3

\displaystyle  = (x-4)(x^2+4x+16)-12x(x-4)

\displaystyle  = (x-4)(x^2-8x+16)

\displaystyle  =(x-4)(x-4)^2 = (x-4)^3

(viii) \displaystyle  x^3+x^2-  \frac{1}{x^2}  +  \frac{1}{x^3} 

\displaystyle  = x^3 +  \frac{1}{x^3}  + x^2 -  \frac{1}{x^2}

\displaystyle  = (x+  \frac{1}{x}  )(x^2 +  \frac{1}{x^2}  -1) + (x-  \frac{1}{x}  )(x+  \frac{1}{x}  )

\displaystyle  = (x +  \frac{1}{x}  )(x^2 +  \frac{1}{x^2}  -1+x -  \frac{1}{x}  )

\displaystyle  = (x +  \frac{1}{x}  ) \Big[ (x -  \frac{1}{x}  ) ((x -  \frac{1}{x}  +1) + 1 \Big]

(ix) \displaystyle  8a^3-b^3-4ax+2bx

\displaystyle  = (2a)^2 - b^3 -4ax+2bx

\displaystyle  = (2a-b)(4a^2+2ab+b^2) - 2x(2a-b)

\displaystyle  = (2a-b)(4a^2+2ab+b^2-2x)

(x) \displaystyle  a^3-  \frac{1}{a^3}  -2a+2  \frac{1}{a}

\displaystyle  = (a -  \frac{1}{a}  )(a^2+1+  \frac{1}{a^2}  )-2(a-  \frac{1}{a}  )

\displaystyle  = (a -  \frac{1}{a}  )(a^2+1+  \frac{1}{a^2}  -2)

\displaystyle  = (a -  \frac{1}{a}  )(a^2+  \frac{1}{a^2}  -1)

(xi) \displaystyle  a^3+b^3+a+b

\displaystyle  = (a+b)(a^2-ab+b^2)+(a+b)

\displaystyle  = (a+b)(a^2-ab+b^2+1)

\displaystyle  \\

Question 6: Factorize:

(i) \displaystyle  64a^3+125b^3+240a^2b+300ab^2

\displaystyle  = (4a)^3+(5b)^3+60ab(4a+5b)

\displaystyle  = (4a+5b)(16a^2-20ab+25b^2)+60ab(4a+5b)

\displaystyle  = (4a+5b)(16a^2-20ab+25b^2 + 60ab)

\displaystyle  = (4a+5b)(16a^2+40ab+25b^2)

\displaystyle  = (4a+5b)(4a+5b)^2 = (4a+5b)^3

(ii) \displaystyle  8x^3+27y^3+36x^2y+54xy^2

\displaystyle  =(2x)^3+(3y)^3+18xy(2x+3y)

\displaystyle  = (2x+3y)(4x^2+9y^2-6xy)+18xy(2x+3y)

\displaystyle  = (2x+3y)(4x^2+9y^2-6xy+18xy)

\displaystyle  = (2x+3y)(4x^2+9y^2+12xy)

\displaystyle  = (2x+3y)(2x+3y)^2 = (2x+3y)^3

(iii) \displaystyle  a^3 - 3a^2b+3ab^2 - b^3 + 8

\displaystyle  = (a-b)^3 +2^3

\displaystyle  = (a-b+2)\Big[ (a-b)^2 -2(a-b)+4 \Big]

\displaystyle  = (a-b+2)(a^2+b^2 - 2ab -2a+2b+4)

(iv) \displaystyle  8x^3+y^3+12x^2y+6xy^2

\displaystyle  = (2x)^3+y^3 + 6xy(2x+y)

\displaystyle  = (2x+y)(4x^2-2xy+y^2) + 6xy(2x+y)

\displaystyle  = (2x+y)(4x^2-2xy+y^2 +6xy)

\displaystyle  = (2x+y)(4x^2+4xy + y^2)

\displaystyle  = (2x+y)(2x+y)^2 = (2x+y)^3

(v) \displaystyle  a^3x^3 - 3a^2bx^2+3ab^2x-b^3

\displaystyle  = (ax)^3 - 3abx(ax-b) - b^3

\displaystyle  = (ax-b)(a^2x^2+abx+b^2) - 3abx(ax-b)

\displaystyle  = (ax-b)(a^2x^2+abx+b^2-3abx)

\displaystyle  = (ax-b)(a^2x^2+b^2-2abx) = (ax-b)(ax-b)^2 = (ax-b)^3

\displaystyle  \\

Question 7: Factorize:

(i) \displaystyle  x^2+12x-45

\displaystyle  = x^2+15x-3x-45

\displaystyle  = x(x+15)-3(x+15)

\displaystyle  = (x+15)(x-3)

(ii) \displaystyle  x^2-22x+120

\displaystyle  = x^2 - 12x - 10x + 120

\displaystyle  = x(x-12)-10(x-12)

\displaystyle  = (x-10)(x-12)

(iii) \displaystyle  x^2-11x-42

\displaystyle  = x^2-14x+3x-42

\displaystyle  = x(x-14)+3(x-14)

\displaystyle  = (x+3)(x-14)

(iv) \displaystyle  y^2+5y-36

\displaystyle  = y^2 +9y-4y-36

\displaystyle  = y(y+9)-4(y+9)

\displaystyle  =(y-4)(y+9)

(v) \displaystyle  (a+b)^2-5(a+b)+4

Let \displaystyle  a+b = x

\displaystyle  = x^2-5x+4

\displaystyle  = x^2-4x-x+ 4

\displaystyle  = x(x-4)-1(x-4)

\displaystyle  = (x-4)(x-1)

\displaystyle  = (a+b-4)(a+b-4)

(vi) \displaystyle  3(x+y)^2-5(x+y)+2

Let \displaystyle  x+y = a

\displaystyle  = 3a^2-5a+2

\displaystyle  = 3a^2 - 3a - 2a + 2

\displaystyle  = 3a(a-1)-2(a-1)

\displaystyle  =(a-1)(3a-2)

\displaystyle  = (x+y-1)(3x+3y-2)

(vii) \displaystyle  (p^2+4p)^2+21(p^2+4p)+98

Let \displaystyle  p^2+4p = x

\displaystyle  = x^2 + 21x + 98

\displaystyle  = x^2 + 14x + 7x + 98

\displaystyle  = x(x+14) + 7( x + 14)

\displaystyle  = (x+14)(x+7)

\displaystyle  = (p^2+4p+14)(p^2+4p+7)

(viii) \displaystyle  x^2-\sqrt{3}x-6

\displaystyle  = x^2 -2\sqrt{3}x+\sqrt{3}x-6

\displaystyle  = x(x+\sqrt{3})-2\sqrt{3}(x+\sqrt{3})

\displaystyle  = (x+\sqrt{3})(x-2\sqrt{3})

(ix) \displaystyle  x^2+5\sqrt{5}x+30

\displaystyle  = x^2 + 3\sqrt{5} x + 2\sqrt{5} x + 3\sqrt{5} \times 2\sqrt{5}

\displaystyle  = x(x+3\sqrt{5}) + 2\sqrt{5}(x+3\sqrt{5})

\displaystyle  = (x+3\sqrt{5})(x+2\sqrt{3})

(x) \displaystyle  (2x^2+5x)(2x^2+5x-19)+84

Let \displaystyle  2x^2+5x=a

\displaystyle  = (a)(a-19)+84

\displaystyle  = a^2-19a+84

\displaystyle  = a^2-12a-7a+84

\displaystyle  = a(a-12)-7(a-12)

\displaystyle  = (a-12)(a-7)

\displaystyle  = (2x^2+5x-12)(2x^2+5x-7)

\displaystyle  = (2x-3)(x+4)(2x+7)(x-1)

\displaystyle  \\

Question 8: Factorize

(i) \displaystyle  5x^2-32x+12

\displaystyle  = 5x^2-30x-2x+12

\displaystyle  = 5x(x-6)-2(x-6)

\displaystyle  = (x-6)(5x-2)

(ii) \displaystyle  30x^2+7x-15

\displaystyle  = 30x^2+25x-18x-15

\displaystyle  = 5x(6x+5)-3(6x+5)

\displaystyle  =(6x+5)(5x-3)

(iii) \displaystyle  6x^2-\sqrt{5}x-5

\displaystyle  = 6x^2-3\sqrt{5}x+2\sqrt{5}x-5

\displaystyle  = 3x(2x-\sqrt{5})+\sqrt{5}(2x-\sqrt{5})

\displaystyle  = (2x-\sqrt{5})(3x+\sqrt{5})

(iv) \displaystyle  \frac{1}{2}  x^2-3x+4

\displaystyle  =  \frac{1}{2}  x^2-2x-x+4

\displaystyle  =  \frac{1}{2}  x(x-4)-1(x-4)

\displaystyle  = (x-4)(  \frac{1}{2}  x -1)

(v) \displaystyle  4\sqrt{3}x^2+5x-2\sqrt{3}

\displaystyle  = 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3}

\displaystyle  =4x(\sqrt{3}x+2) - \sqrt{3}(\sqrt{3}x+2)

\displaystyle  = (\sqrt{3}x+2)(4x-\sqrt{3})

(vi) \displaystyle  \frac{a}{b}  x^2+(  \frac{a}{b}  +  \frac{c}{d})  x +  \frac{c}{d}

\displaystyle  =  \frac{a}{b}  x (x+1) +  \frac{c}{d}  (x+1)

\displaystyle  = (x+1) (  \frac{a}{b}  x+  \frac{c}{d}  )

(vii) \displaystyle  px^2+(4p^2-3q)x-12pq

\displaystyle  = px^2 + 4p^2x-3qx-12pq

\displaystyle  = px(x+4p)-3q(x+4p)

\displaystyle  = (x+4p)(px-3q)

(viii) \displaystyle  3(a-2)^2-2(a-2)-8

Let \displaystyle  a-2 = x

\displaystyle  = 3x^2-2x-8

\displaystyle  = 3x^2 - 6x+4x-8

\displaystyle  = 3x(x-2)+4(x-2)

\displaystyle  = (x-2)(3x+4)

(ix) \displaystyle  12(a+1)^2-25(a+1)(b+2)+12(b+2)^2

Let \displaystyle  a+1 = x and \displaystyle  b+2 = y

\displaystyle  = 12x^2 - 25xy + 12y^2

\displaystyle  = 12x^2 - 9xy - 16xy + 12y^2

\displaystyle  = 3x(4x-3y) + 4y(4x-3y)

\displaystyle  = (4x-3y)(3x+4y)

\displaystyle  = \Big( 4(a+1)-3(b+2) \Big) \Big( 3(a+1)+4(b+2) \Big)

\displaystyle  = (4a-3b-2)(3a-4b-5)

(x) \displaystyle  5x^6-7x^3-6

Let \displaystyle  x^3 = a

\displaystyle  = 5a^2-7a-6

\displaystyle  = 5a^2-10a+3a-6

\displaystyle  = 5a^2(a-2)+3(a-2)

\displaystyle  = (a-2)(5a+3)

\displaystyle  =(x^3-2)(5x^3+3)

(xi) \displaystyle  x^2+  \frac{12}{35}  x+  \frac{1}{35}

\displaystyle  = x^2 +  \frac{1}{7}  x +  \frac{1}{5}  x +  \frac{1}{35}

\displaystyle  = x(x+  \frac{1}{7}  )+  \frac{1}{5}  (x+  \frac{1}{7}  )

\displaystyle  = (x+  \frac{1}{7}  )(x+  \frac{1}{5}  )

(xii) \displaystyle  2x^2+3\sqrt{3}x+3

\displaystyle  = 2x^2 + 2\sqrt{3} x + \sqrt{3} x + 3

\displaystyle  = 2x(x+\sqrt{3}) + \sqrt{3}(x+\sqrt{3})

\displaystyle  = (x+\sqrt{3})(2x+\sqrt{3})

(xiii) \displaystyle  5\sqrt{5}x^2+20x+3\sqrt{5}

\displaystyle  = 5\sqrt{5} x^2 + 15x + 5 x + 3\sqrt{5}

\displaystyle  = \sqrt{5}x(5x+\sqrt{5}) + 3 (5x+\sqrt{5})

\displaystyle  = (5x+\sqrt{5})(\sqrt{5}x+3)

(xiv) \displaystyle  2x^2+3\sqrt{5}x+5

\displaystyle  = 2x^2+2\sqrt{5}x+\sqrt{5}x+5

\displaystyle  = 2x(x+\sqrt{5}) + \sqrt{5}(x+\sqrt{5})

\displaystyle  = (x+\sqrt{5})(2x+\sqrt{5})

(xv) \displaystyle  7x^2+2\sqrt{14}x+2

\displaystyle  = 7x^2+\sqrt{14} x + \sqrt{14} x + 2

\displaystyle  = 7x^2+\sqrt{2 \times 7} x + \sqrt{2 \times 7} x + 2

\displaystyle  = \sqrt{7}x(\sqrt{7}x+\sqrt{2})+\sqrt{2}(\sqrt{7}x+\sqrt{2})

\displaystyle  = (\sqrt{7}x + \sqrt{2})(\sqrt{7}x+\sqrt{2})

\displaystyle  = (\sqrt{7}x + \sqrt{2})^2