\displaystyle \textbf{Question 1: }\text{Factorize }x^2-81.
\displaystyle \text{Answer:}
\displaystyle x^2-81=x^2-9^2
\displaystyle =(x-9)(x+9).
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Factorize }9a^2-25.
\displaystyle \text{Answer:}
\displaystyle 9a^2-25=(3a)^2-5^2
\displaystyle =(3a-5)(3a+5).
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Factorize }36y^2-121.
\displaystyle \text{Answer:}
\displaystyle 36y^2-121=(6y)^2-11^2
\displaystyle =(6y-11)(6y+11).
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Factorize }49a^2-100b^2.
\displaystyle \text{Answer:}
\displaystyle 49a^2-100b^2=(7a)^2-(10b)^2
\displaystyle =(7a-10b)(7a+10b).
\displaystyle \\

\displaystyle \textbf{Question 5: }\text{Factorize }(a+b)^2-36.
\displaystyle \text{Answer:}
\displaystyle (a+b)^2-36=(a+b)^2-6^2
\displaystyle =(a+b-6)(a+b+6).
\displaystyle \\

\displaystyle \textbf{Question 6: }\text{Factorize }16c^2-1.
\displaystyle \text{Answer:}
\displaystyle 16c^2-1=(4c)^2-1^2
\displaystyle =(4c-1)(4c+1).
\displaystyle \\

\displaystyle \textbf{Question 7: }\text{Factorize }1-64b^2.
\displaystyle \text{Answer:}
\displaystyle 1-64b^2=1^2-(8b)^2
\displaystyle =(1-8b)(1+8b).
\displaystyle \\

\displaystyle \textbf{Question 8: }\text{Factorize } \frac{9}{16}-25x^2.
\displaystyle \text{Answer:}
\displaystyle \frac{9}{16}-25x^2 =\left(\frac{3}{4}\right)^2-(5x)^2
\displaystyle =\left(\frac{3}{4}-5x\right)\left(\frac{3}{4}+5x\right).
\displaystyle \\

\displaystyle \textbf{Question 9: }\text{Factorize } z^2-\frac{1}{144}.
\displaystyle \text{Answer:}
\displaystyle z^2-\frac{1}{144} =z^2-\left(\frac{1}{12}\right)^2
\displaystyle =\left(z-\frac{1}{12}\right)\left(z+\frac{1}{12}\right).
\displaystyle \\

\displaystyle \textbf{Question 10: }\text{Factorize } 1-(a-b)^2.
\displaystyle \text{Answer:}
\displaystyle 1-(a-b)^2 =\{1-(a-b)\}\{1+(a-b)\}
\displaystyle =(1-a+b)(1+a-b).
\displaystyle \\

\displaystyle \textbf{Question 11: }\text{Factorize } (3m-n)^2-(m-2n)^2.
\displaystyle \text{Answer:}
\displaystyle (3m-n)^2-(m-2n)^2
\displaystyle =\{(3m-n)-(m-2n)\}\{(3m-n)+(m-2n)\}
\displaystyle =(3m-n-m+2n)(3m-n+m-2n)
\displaystyle =(2m+n)(4m-3n).
\displaystyle \\

\displaystyle \textbf{Question 12: }\text{Factorize } (3x+2y)^2-(2x-3y)^2.
\displaystyle \text{Answer:}
\displaystyle (3x+2y)^2-(2x-3y)^2
\displaystyle =\{(3x+2y)+(2x-3y)\}\{(3x+2y)-(2x-3y)\}
\displaystyle =(5x-y)(x+5y).
\displaystyle \\

\displaystyle \textbf{Question 13: }\text{Factorize } 16(a+b)^2-9(a-b)^2.
\displaystyle \text{Answer:}
\displaystyle 16(a+b)^2-9(a-b)^2
\displaystyle =\{4(a+b)-3(a-b)\}\{4(a+b)+3(a-b)\}
\displaystyle =(4a+4b-3a+3b)(4a+4b+3a-3b)
\displaystyle =(a+7b)(7a+b).
\displaystyle \\

\displaystyle \textbf{Question 14: }\text{Factorize } 9(x+y)^2-16(x-2y)^2.
\displaystyle \text{Answer:}
\displaystyle 9(x+y)^2-16(x-2y)^2
\displaystyle =\{3(x+y)-4(x-2y)\}\{3(x+y)+4(x-2y)\}
\displaystyle =(3x+3y-4x+8y)(3x+3y+4x-8y)
\displaystyle =(11y-x)(7x-5y).
\displaystyle \\

\displaystyle \textbf{Question 15: }\text{Factorize } 36(a-b)^2-25(a+b)^2.
\displaystyle \text{Answer:}
\displaystyle 36(a-b)^2-25(a+b)^2
\displaystyle =\{6(a-b)-5(a+b)\}\{6(a-b)+5(a+b)\}
\displaystyle =(6a-6b-5a-5b)(6a-6b+5a+5b)
\displaystyle =(a-11b)(11a-b).
\displaystyle \\

\displaystyle \textbf{Question 16: }\text{Factorize } 9(3x+1)^2-4(x-1)^2.
\displaystyle \text{Answer:}
\displaystyle 9(3x+1)^2-4(x-1)^2
\displaystyle =\{3(3x+1)-2(x-1)\}\{3(3x+1)+2(x-1)\}
\displaystyle =(9x+3-2x+2)(9x+3+2x-2)
\displaystyle =(7x+5)(11x+1).
\displaystyle \\

\displaystyle \textbf{Question 17: }\text{Factorize } a^2-2ab+b^2-c^2.
\displaystyle \text{Answer:}
\displaystyle a^2-2ab+b^2-c^2
\displaystyle =(a-b)^2-c^2
\displaystyle =(a-b-c)(a-b+c).
\displaystyle \\

\displaystyle \textbf{Question 18: }\text{Factorize } x^2-a^2-2a-1.
\displaystyle \text{Answer:}
\displaystyle x^2-a^2-2a-1
\displaystyle =x^2-(a^2+2a+1)
\displaystyle =x^2-(a+1)^2
\displaystyle =(x-a-1)(x+a+1).
\displaystyle \\

\displaystyle \textbf{Question 19: }\text{Factorize } x^2-m^2+6mn-9n^2.
\displaystyle \text{Answer:}
\displaystyle x^2-m^2+6mn-9n^2
\displaystyle =x^2-(m^2-6mn+9n^2)
\displaystyle =x^2-(m-3n)^2
\displaystyle =\{x-(m-3n)\}\{x+(m-3n)\}
\displaystyle =(x-m+3n)(x+m-3n).
\displaystyle \\

\displaystyle \textbf{Question 20: }\text{Factorize }a^4-b^4.
\displaystyle \text{Answer:}
\displaystyle a^4-b^4=(a^2)^2-(b^2)^2
\displaystyle =(a^2-b^2)(a^2+b^2)
\displaystyle =(a-b)(a+b)(a^2+b^2).
\displaystyle \\

\displaystyle \textbf{Question 21: }\text{Factorize }16a^4-81b^4.
\displaystyle \text{Answer:}
\displaystyle 16a^4-81b^4=(4a^2)^2-(9b^2)^2
\displaystyle =(4a^2-9b^2)(4a^2+9b^2)
\displaystyle =(2a-3b)(2a+3b)(4a^2+9b^2).
\displaystyle \\

\displaystyle \textbf{Question 22: }\text{Factorize }3-75z^2.
\displaystyle \text{Answer:}
\displaystyle 3-75z^2=3(1-25z^2)
\displaystyle =3\{1^2-(5z)^2\}
\displaystyle =3(1-5z)(1+5z).
\displaystyle \\

\displaystyle \textbf{Question 23: }\text{Factorize }48a^2b^2-3.
\displaystyle \text{Answer:}
\displaystyle 48a^2b^2-3=3(16a^2b^2-1)
\displaystyle =3\{(4ab)^2-1^2\}
\displaystyle =3(4ab-1)(4ab+1).
\displaystyle \\

\displaystyle \textbf{Question 24: }\text{Factorize }4x^3-81x.
\displaystyle \text{Answer:}
\displaystyle 4x^3-81x=x(4x^2-81)
\displaystyle =x\{(2x)^2-9^2\}
\displaystyle =x(2x-9)(2x+9).
\displaystyle \\

\displaystyle \textbf{Question 25: }\text{Factorize }9b^3-144b.
\displaystyle \text{Answer:}
\displaystyle 9b^3-144b=9b(b^2-16)
\displaystyle =9b(b^2-4^2)
\displaystyle =9b(b-4)(b+4).
\displaystyle \\

\displaystyle \textbf{Question 26: }\text{Factorize }32x^2-72y^2.
\displaystyle \text{Answer:}
\displaystyle 32x^2-72y^2=8(4x^2-9y^2)
\displaystyle =8\{(2x)^2-(3y)^2\}
\displaystyle =8(2x-3y)(2x+3y).
\displaystyle \\

\displaystyle \textbf{Question 27: }\text{Factorize }50x^2-32y^2.
\displaystyle \text{Answer:}
\displaystyle 50x^2-32y^2=2(25x^2-16y^2)
\displaystyle =2\{(5x)^2-(4y)^2\}
\displaystyle =2(5x-4y)(5x+4y).
\displaystyle \\

\displaystyle \textbf{Question 28: }\text{Factorize }a^3-4ab^2.
\displaystyle \text{Answer:}
\displaystyle a^3-4ab^2=a(a^2-4b^2)
\displaystyle =a\{a^2-(2b)^2\}
\displaystyle =a(a-2b)(a+2b).
\displaystyle \\

\displaystyle \textbf{Question 29: }\text{Factorize }ab^3c-abc^3.
\displaystyle \text{Answer:}
\displaystyle ab^3c-abc^3=abc(b^2-c^2)
\displaystyle =abc(b-c)(b+c).
\displaystyle \\

\displaystyle \textbf{Question 30: }\text{Factorize } 9(x+y)^3-16(x+y).
\displaystyle \text{Answer:}
\displaystyle 9(x+y)^3-16(x+y)
\displaystyle =(x+y)\{9(x+y)^2-16\}
\displaystyle =(x+y)\{[3(x+y)]^2-4^2\}
\displaystyle =(x+y)\{3(x+y)-4\}\{3(x+y)+4\}
\displaystyle =(x+y)(3x+3y-4)(3x+3y+4).
\displaystyle \\

\displaystyle \textbf{Question 31: }\text{Factorize }1-0.49c^6.
\displaystyle \text{Answer:}
\displaystyle 1-0.49c^6=1^2-(0.7c^3)^2
\displaystyle =(1-0.7c^3)(1+0.7c^3).
\displaystyle \\

\displaystyle \textbf{Question 32: }\text{Factorize } x^2-y^2-8yz-16z^2.
\displaystyle \text{Answer:}
\displaystyle x^2-y^2-8yz-16z^2
\displaystyle =x^2-(y^2+8yz+16z^2)
\displaystyle =x^2-(y+4z)^2
\displaystyle =(x-y-4z)(x+y+4z).
\displaystyle \\

\displaystyle \textbf{Question 33: }\text{Factorize } x^3y^3-\frac{25xy}{z^2}.
\displaystyle \text{Answer:}
\displaystyle x^3y^3-\frac{25xy}{z^2} =xy\left(x^2y^2-\frac{25}{z^2}\right)
\displaystyle =xy\left\{(xy)^2-\left(\frac{5}{z}\right)^2\right\}
\displaystyle =xy\left(xy-\frac{5}{z}\right)\left(xy+\frac{5}{z}\right).
\displaystyle \\

\displaystyle \textbf{Question 34: }\text{Factorize } 324x^4-0.0064b^4.
\displaystyle \text{Answer:}
\displaystyle 324x^4-0.0064b^4 =(18x^2)^2-(0.08b^2)^2
\displaystyle =(18x^2-0.08b^2)(18x^2+0.08b^2)
\displaystyle =2(9x^2-0.04b^2)(18x^2+0.08b^2)
\displaystyle =2(3x-0.2b)(3x+0.2b)(18x^2+0.08b^2).
\displaystyle \\

\displaystyle \textbf{Question 35: }\text{Using the identity } a^2-b^2=(a+b)(a-b),\text{ evaluate the following:}
\displaystyle \text{(i) }82^2-18^2
\displaystyle \text{(ii) }8^2-9.2^2
\displaystyle \text{(iii) }8^2-0.2^2
\displaystyle \text{(iv) }\left(7\frac{3}{4}\right)^2- \left(2\frac{1}{4}\right)^2
\displaystyle \text{(v) }\left(6\frac{4}{11}\right)^2- \left(4\frac{7}{11}\right)^2
\displaystyle \text{(vi) } \frac{7.3\times7.3-2.7\times2.7}{7.3-2.7}
\displaystyle \text{Answer:}

\displaystyle \text{(i) }82^2-18^2=(82+18)(82-18)
\displaystyle =100\times64=6400.

\displaystyle \text{(ii) }8^2-9.2^2=(8+9.2)(8-9.2)
\displaystyle =17.2\times(-1.2)=-20.64.

\displaystyle \text{(iii) }8^2-0.2^2=(8+0.2)(8-0.2)
\displaystyle =8.2\times7.8=63.96.

\displaystyle \text{(iv) }\left(7\frac{3}{4}\right)^2- \left(2\frac{1}{4}\right)^2
\displaystyle =\left(7\frac{3}{4}+2\frac{1}{4}\right) \left(7\frac{3}{4}-2\frac{1}{4}\right)
\displaystyle =10\times5\frac{1}{2}=55.

\displaystyle \text{(v) }\left(6\frac{4}{11}\right)^2- \left(4\frac{7}{11}\right)^2
\displaystyle =\left(6\frac{4}{11}+4\frac{7}{11}\right) \left(6\frac{4}{11}-4\frac{7}{11}\right)
\displaystyle =11\times1\frac{8}{11}
\displaystyle =11\times\frac{19}{11}=19.

\displaystyle \text{(vi) } \frac{7.3\times7.3-2.7\times2.7}{7.3-2.7}
\displaystyle =\frac{(7.3+2.7)(7.3-2.7)}{7.3-2.7}
\displaystyle =7.3+2.7=10.
\displaystyle \\

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