Further problems 1

Exercise 1

Simplify the terms:

  1. 30x+(5x−2y)30x + (5x-2y)

  2. (2a−2b)+(3a+4b)+(5a−6b)(2a-2b)+(3a+4b)+(5a-6b)

  3. 6a−2b+5c−(−7b+4c)6a-2b+5c-(-7b+4c)

  4. 16a−(3b+8c−5a)−(b−3c)16a-(3b+8c-5a)-(b-3c)

  5. 25a−[36b−(19a−11b)−12a]25a-[36b-(19a-11b)-12a]

Solution
  1. 30x+(5x−2y)=35x−2y‾30x + (5x-2y) = \underline{35x-2y}
  2. (2a−2b)+(3a+4b)+(5a−6b)=10a−4b‾(2a-2b)+(3a+4b)+(5a-6b) = \underline{10a-4b}
  3. 6a−2b+5c−(−7b+4c)=6a−2b+5c+7b−4c=6a+5b+c‾6a-2b+5c-(-7b+4c)= 6a-2b+5c+7b-4c = \underline{6a+5b+c}
  4. 16a−(3b+8c−5a)−(b−3c)=16a−3b−8c+5a−b+3c=21a−4b−5c‾16a-(3b+8c-5a)-(b-3c) = 16a-3b-8c+5a-b+3c = \underline{21a-4b-5c}
  5. 25a−[36b−(19a−11b)−12a]=25a−36b+(19a−11b)+12a=56a−47b‾25a-[36b-(19a-11b)-12a]=25a-36b+(19a-11b)+12a =\underline{56a-47b}
Exercise 2

Expand as much as possible, and simplify:

  1. 4(a+b)4(a+b)

  2. 4a(3b−5c)4a(3b-5c)

  3. (a+b−c)⋅(−34)(a+b-c)\cdot (-\frac{3}{4})

  4. (m−n)(x+y)(m-n)(x+y)

  5. (a−5m)(a+3n)(a-5m)(a+3n)

  6. (−8x+5y)(2x−3y)(-8x+5y)(2x-3y)

  7. (2x+y)(3m+n)+(2x+y)(m−3n)(2x+y)(3m+n)+(2x+y)(m-3n)

  8. (a+b)(4x−5y)−(a−b)(5x+3y)(a+b)(4x-5y)-(a-b)(5x+3y)

  9. (4y+6x)(3a−5b)−(2x−6y)(2a+3b)(4y+6x)(3a-5b)-(2x-6y)(2a+3b)

  10. (2a−6c)(4b−2d)−(a+c)(5b−3d)(−3x)(2a-6c)(4b-2d) -(a+c)(5b-3d)(-3x)

  11. (7x+5)2(7x+5)^2

  12. (x−1)2(x-1)^2

  13. (n+2n)2(n+\frac{2}{n})^2

  14. (a−1)(a+1)(a-1)(a+1)

  15. (x+y−1)(x−y+1)(x+y-1)(x-y+1)

  16. (2−a)2−(a+2)(2−a)+(a+2)2(2-a)^2-(a+2)(2-a)+(a+2)^2

  17. 6ax(6d−c)(4b+4y)3n6ax(6d-c)(4b+4y)3n

Solution
  1. 4(a+b)=4a+4b‾4(a+b) = \underline{4a + 4b}
  2. 4a(3b−5c)=12ab−20ac‾4a(3b-5c) = \underline{12ab-20ac}
  3. (a+b−c)⋅(−34)=−34a−34b+34c‾(a+b-c)\cdot (-\frac{3}{4}) = \underline{-\frac{3}{4}a-\frac{3}{4}b+\frac{3}{4}c}
  4. (m−n)(x+y)=mx+my−nx−ny‾(m-n)(x+y) = \underline{mx+my-nx-ny}
  5. (a−5m)(a+3n)=a2+3an−5am−15mn‾(a-5m)(a+3n)= \underline{a^2+3an-5am-15mn}
  6. (−8x+5y)(2x−3y)=−16x2+24xy+10xy−15y2=−16x2+34xy−15y2‾(-8x+5y)(2x-3y) = -16x^2+24xy+10xy-15y^2 = \underline{-16x^2+34xy-15y^2}
  7. (2x+y)(3m+n)+(2x+y)(m−3n)=8mx−4nx+4my−2ny‾(2x+y)(3m+n)+(2x+y)(m-3n)=\underline{8mx-4nx+4my-2ny}
  8. (a+b)(4x−5y)−(a−b)(5x+3y)=−ax−8ay+9bx−2by‾(a+b)(4x-5y)-(a-b)(5x+3y)=\underline{-ax-8ay+9bx-2by}
  9. (4y+6x)(3a−5b)−(2x−6y)(2a+3b)=24ay−2by+14ax−36bx‾(4y+6x)(3a-5b)-(2x-6y)(2a+3b)= \underline{24ay-2by+14ax-36bx}
  10. (2a−6c)(4b−2d)−(a+c)(5b−3d)(−3x)=8ab−4ad−24bc+12cd+15abx−9adx+15bcx−9cdx‾(2a-6c)(4b-2d) -(a+c)(5b-3d)(-3x) = \underline{8ab-4ad-24bc+12cd + 15abx-9adx+15bcx-9cdx}
  11. (7x+5)2=49x2+70x+25‾(7x+5)^2 = \underline{49x^2+70x+25}
  12. (x−1)2=x2−2x+1‾(x-1)^2= \underline{x^2-2x+1}
  13. (n+2n)2=n2+2n2n+(2n)2=n2+4+4n2‾(n+\frac{2}{n})^2 =n^2+2n\frac{2}{n}+\left(\frac{2}{n}\right)^2= \underline{n^2+4+\frac{4}{n^2}}
  14. (a−1)(a+1)=a2−1‾(a-1)(a+1) = \underline{a^2-1}
  15. (x+y−1)(x−y+1)=x2−y2+2y−1‾(x+y-1)(x-y+1)= \underline{x^2 - y^2 + 2y -1}
  16. (2−a)2−(a+2)(2−a)+(a+2)2=3a2+4‾(2-a)^2-(a+2)(2-a)+(a+2)^2 = \underline{3a^2+4}
  17. 6ax(6d−c)(4b+4y)3n=432abdnx+432adnxy−72abcnx−72acnxy‾6ax(6d-c)(4b+4y)3n = \underline{432a b d n x + 432a d n x y-72 a b c n x -72a c n x y}
Exercise 3

Factor out and simplify:

  1. ab+ac+ad−aeab+ac+ad-ae

  2. 2(x−3)+y(x−3)2(x-3)+y(x-3)

  3. (5m+2n)(x−y)+(3m+2n)(x−y)(5m+2n)(x-y)+(3m+2n)(x-y)

  4. a(x+y)−x−ya(x+y)-x-y

  5. c2−2cd+d2c^2-2cd+d^2

  6. 25x2−70x+4925x^2-70x+49

  7. 36a2b2−49c236a^2b^2-49c^2

  8. 0.49bc2−81bd20.49bc^2-81bd^2

  9. (a2−2)2−(a+1)2(a^2-2)^2-(a+1)^2

  10. x2−6x+9−y2x^2-6x+9-y^2

Solution
  1. ab+ac+ad−ae=a(b+c+d−e)‾ab+ac+ad-ae = \underline{a(b+c+d-e)}
  2. 2(x−3)+y(x−3)=(2+y)(x−3)‾2(x-3)+y(x-3)= \underline{(2+y)(x-3)}
  3. (5m+2n)(x−y)+(3m+2n)(x−y)=(5m+2n+3m+2n)(x−y)=(8m+4n)(x−y)=4(2m+n)(x−y)‾(5m+2n)(x-y)+(3m+2n)(x-y) = (5m+2n+3m+2n)(x-y)= (8m+4n)(x-y) = \underline{4(2m+n)(x-y)}
  4. a(x+y)−x−y=a(x+y)−(x+y)=(a−1)(x+y)‾a(x+y)-x-y= a(x+y) - (x+y)= \underline{(a-1)(x+y)}
  5. c2−2cd+d2=(c−d)2‾c^2-2cd+d^2 = \underline{(c-d)^2}
  6. 25x2−70x+49=(5x−7)2‾25x^2-70x+49 = \underline{(5x-7)^2}
  7. 36a2b2−49c2=(6ab)2−(7c)2=(6ab+7c)(6ab−7c)‾36a^2b^2-49c^2= (6ab)^2-(7c)^2 = \underline{(6ab+7c)(6ab-7c)}
  8. 0.49bc2−81bd2=b[(0.7c)2−(9d)2]=b(0.7c+9d)(0.7c−9d)‾0.49bc^2-81bd^2 = b[(0.7c)^2-(9d)^2]= \underline{b(0.7c+9d)(0.7c-9d)}
  9. (a2−2)2−(a+1)2=(a2−2+a+1)(a2−2−a−1)=(a2+a−1)(a2−a−3)‾(a^2-2)^2-(a+1)^2 = (a^2-2+a+1)(a^2-2-a-1) = \underline{(a^2+a-1)(a^2-a-3)}
  10. x2−6x+9−y2=(x−3)2−y2=(x−3+y)(x−3−y)‾x^2-6x+9-y^2= (x-3)^2-y^2 = \underline{(x-3+y)(x-3-y)}