Factorize

Question 1:  x^2-81

Answer:

x^2-81 = (x-9)(x+9)

\\

Question 2:  9a^2-25

Answer:

9a^2-25 = (3a-5)(3a+5)

\\

Question 3:  36y^2-121 

Answer:

36y^2-121 = (6y-11)(6y+11)

\\

Question 4:  49a^2-100b^2

Answer:

49a^2-100b^2 = (7a-10b)(7a+10b)

\\

Question 5:  (a+b^2 )-36

Answer:

(a+b^2 )-36 = (a+b-6)(a+b+6)

\\

Question 6:  16c^2-1 

Answer:

16c^2-1 = (4c-1)(4c+1)

\\

Question 7:  1-64b^2 

Answer:

1-64b^2 = (1-8b)(1+8b)

\\

Question 8:  \frac{9}{16} -25x^2 

Answer:

\frac{9}{16} -25x^2 = \Big( \frac{3}{4} -5x \Big) \Big( \frac{3}{4} +5x \Big)

\\

Question 9:  z^2- \frac{1}{144} 

Answer:

z^2- \frac{1}{144} = \Big(z- \frac{1}{12} \Big) \Big(z+ \frac{1}{12} \Big)

\\

Question 10:  1-(a-b)^2 

Answer:

1-(a-b)^2 = (1-a+b)(1+a-b)

\\

Question 11:  (3m-n)^2-(m-2n)^2

Answer:

(3m-n)^2-(m-2n)^2

= [(3m-n)-(m-2n)][(3m-n)+(m-2n)]

= (3m-n-m+2n)(3m-n+m-2n)

= (2m+n)(4m-3n)

\\

Question 12:  (3x+2y)^2-(2x-3y)^2

Answer:

(3x+2y)^2-(2x-3y)^2

= [(3x+2y)+(2x-3y)][(3x+2y)-(2x-3y)]

= (5x-y)(x+5y)

\\

Question 13:  16(a+b)^2-9(a-b)^2

Answer:

16(a+b)^2-9(a-b)^2

= [4(a+b)+3(a-b)][4(a+b)+3(a-b)]

= (a+7b)(7a+b)

\\

Question 14:  9(x+y)^2-16(x-2y)^2

Answer:

9(x+y)^2-16(x-2y)^2

= [3(x+y)-4(x-2y)][3(x+y)+4(x-2y)]

= (11y-x)(7x-5y)

\\

Question 15: 36(a-b)^2-25(a+b)^2

Answer:

36(a-b)^2-25(a+b)^2

= [6(a-b)-5(a+b)][6(a-b)+5(a+b)]

= (6a-6b-5a-5b)(6a-6b+5a+5b)

= (a-11b)(11a-b)

\\

Question 16:  9(3x+1)^2-4(x-1)^2

Answer:

9(3x+1)^2-4(x-1)^2

= [3(3x+1)-2(x-1)][3(3x+1)+(x-1)]

= (9x+3-2x+2)(9x+3+2x-2)

= (7x+5)(11x+1)

\\

Question 17:  a^2-2ab+b^2-c^2

Answer:

a^2-2ab+b^2-c^2   = (a-b)^2-c^2   = (a-b-c)(a-b+c)

\\

Question 18:  x^2-a^2-2a-1

Answer:

x^2-a^2-2a-1

= x^2-(a^2+2a+1)

= x^2-(a+1)^2

= (x-a-1)(x+a+1)

\\

Question 19:  x^2-m^2+6mn-9n^2

Answer:

x^2-m^2+6mn-9n^2

= x^2-(m^2-6mn+9n^2)

= x^2-(m-3n)^2

= (x-m+3n)(x+m-3n)

\\

Question 20:  a^4-b^4

Answer:

a^4-b^4   = (a^2+b^2)(a^2-b^2)   = (a^2+b^2 )(a-b)(a+b)

\\

Question 21: 16a^4-81b^4

Answer:

16a^4-81b^4   = (4a^2-9b^2)(4a^2+9b^2)   = (2a-3b)(2a+3b)(4a^2+9b^2 )

\\

Question 22: 3-75z^2

Answer:

3-75z^2   = 3(1-25z^2 )   = 3(1-5z)(1+5z)

\\

Question 23: 48a^2 b^2-3

Answer:

48a^2 b^2-3   = 3(16a^2 b^2-1)   = 3(4ab-1)(4ab+1)

\\

Question 24: 4x^3-81x

Answer:

4x^3-81x   = x(4x^2-81)   = x(2x-9)(2x+9)

\\

Question 25: 9b^3-144b

Answer:

9b^3-144b   = b(9b^2-144)   = b(3b-12)(3b+12)

\\

Question 26: 32x^2-72y^2

Answer:

32x^2-72y^2   = 2(16x^2-36y^2 )   = 2(4x-6y)(4x-6y)

\\

Question 27: 50x^2-32y^2

Answer:

50x^2-32y^2   = 2y(25x^2-16y^2 )   = 2y(5x-4y)(5x+4y)

\\

Question 28: a^3-4ab^2

Answer:

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

\\

Question 29: ab^3 c-abc^3

Answer:

ab^3 c-abc^3   = abc(b^2-c^2 )   = abc(b-c)(b+c)

\\

Question 30: 9(x+y)^3-16(x+y)

Answer:

9(x+y)^3-16(x+y)

= (x+y)[9(x+y)^2-16]

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

\\

Question 31: 1-0.49c^6

Answer:

1-0.49c^6   = (1-0.7c^3 )(1+0.7c^3 )

\\

Question 32: x^2-y^2-8yz-16z^2

Answer:

x^2-y^2-8yz-16z^2

= x^2-(y^2+8yz+16z^2 )

= x^2-(y+4z)^2

= (x-y-4z)(x+y+4z)

\\

Question 33: x^3 y^3- \frac{25xy}{z^2}

Answer:

x^3 y^3- \frac{25xy}{z^2} = xy \Big(x^2 y^2- \frac{25}{z^2} \Big)   = xy \Big( xy+ \frac{5}{2} \Big) \Big( xy- \frac{5}{2} \Big)

\\

Question 34: 324x^4-0.0064b^4

Answer:

324x^4-0.0064b^4

= 1/1000 (324x^4-64b^4 )

= 4/100 (81x^4-16b^4 )

= 4/100 (9x^2-4b^2 )(9x^2+4b^2 )

= 0.0004(3x-2b)(3x+2b)(9x^2+4b^2 )

= (0.09x^2+0.04b^2 )(0.3x+0.2b)(0.3x-0.2b)

\\

Question 35:  Using the identity (a^2 - b^2 ) = (a+b)(a-b) , evaluate each of the following:

i)  82^2-18^2       ii)  8^2-9.2^2       iii)  8^2-0.2^2 

iv)  (7 \frac{3}{4})^2-(2 \frac{1}{4})^2       v)  (6 \frac{4}{11})^2-(4 \frac{7}{11})^2       vi)  \frac{(7.3 \times 7.3-2.7 \times 2.7)}{(7.3-2.7)}

Answer:

i)  82^2-18^2 = (82+18)(82-18) = 100 \times 64 = 6400

ii)  8^2-9.2^2 = (15.8+9.2)(15.8-9.2) = 25 \times 6.6 = 165

iii)  8^2-0.2^2 = (0.8+0.2)(0.8-0.2) = 1 \times (.06) = 0.6

iv)  \Big(7 \frac{3}{4} \Big)^2-(2 \frac{1}{4} \Big)^2 = \Big(7 \frac{3}{4} +2 \frac{1}{4} \Big) \Big(7 \frac{3}{4} -2 \frac{1}{4} \Big) = 10 \times 5 \frac{1}{2} = 55

v)  \Big(6 \frac{4}{11} \Big)^2-\Big(4 \frac{7}{11} \Big)^2 = \Big(6 \frac{4}{11} +4 \frac{7}{11} \Big) \Big(6 \frac{4}{11} -4 \frac{7}{11} \Big) = 11 \times 1 \frac{4}{11} = 19

vi) \frac{(7.3 \times 7.3-2.7 \times 2.7)}{(7.3-2.7)} = \frac{(7.3+2.7)(7.3-2.7)}{(7.3-2.7)} = 10