Class 9: Factorization

$\displaystyle \textbf{Exercise 5(A)}$

$\displaystyle \textbf{Question 1: }\text{Factorise: }3a^2-9ab$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The H.C.F. of }3a^2\text{ and }9ab\text{ is }3a.$
$\displaystyle \therefore 3a^2-9ab=3a(a-3b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Factorise: }2(x+y)^3-6(x+y)$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The common factor is }2(x+y).$
$\displaystyle \therefore 2(x+y)^3-6(x+y)=2(x+y)\left[(x+y)^2-3\right]$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Factorise: }x^3(2x-3y)-x^2(2x-3y)^2$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The common factor is }x^2(2x-3y).$
$\displaystyle \therefore x^3(2x-3y)-x^2(2x-3y)^2=x^2(2x-3y)\left[x-(2x-3y)\right]$
$\displaystyle =x^2(2x-3y)(3y-x)$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Factorise: }2(2x-5y)(3x+4y)-6(2x-5y)(x-y)$
$\displaystyle \text{Answer:}$
$\displaystyle \text{The common factor is }2(2x-5y).$
$\displaystyle =2(2x-5y)\left[(3x+4y)-3(x-y)\right]$
$\displaystyle =2(2x-5y)(3x+4y-3x+3y)$
$\displaystyle =2(2x-5y)(7y)$
$\displaystyle \therefore 2(2x-5y)(3x+4y)-6(2x-5y)(x-y)=14y(2x-5y)$

$\displaystyle \textbf{Question 5: }\text{Factorise using the grouping method: }a^3+a-3a^2-3$
$\displaystyle \text{Answer:}$
$\displaystyle a^3+a-3a^2-3=a^3-3a^2+a-3$
$\displaystyle =a^2(a-3)+1(a-3)$
$\displaystyle =(a-3)(a^2+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Factorise using the grouping method: }16(a+b)^2-4a-4b$
$\displaystyle \text{Answer:}$
$\displaystyle 16(a+b)^2-4a-4b=16(a+b)^2-4(a+b)$
$\displaystyle =4(a+b)\left[4(a+b)-1\right]$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{Factorise using the grouping method: }a^4-2a^3-4a+8$
$\displaystyle \text{Answer:}$
$\displaystyle a^4-2a^3-4a+8=a^3(a-2)-4(a-2)$
$\displaystyle =(a-2)(a^3-4)$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Factorise using the grouping method: }ab-2b+a^2-2a$
$\displaystyle \text{Answer:}$
$\displaystyle ab-2b+a^2-2a=b(a-2)+a(a-2)$
$\displaystyle =(a-2)(a+b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Factorise using the grouping method: }ab(x^2+1)+x(a^2+b^2)$
$\displaystyle \text{Answer:}$
$\displaystyle ab(x^2+1)+x(a^2+b^2)=abx^2+ab+a^2x+b^2x$
$\displaystyle =(abx^2+a^2x)+(ab+b^2x)$
$\displaystyle =ax(bx+a)+b(a+bx)$
$\displaystyle =(a+bx)(ax+b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Factorise using the grouping method: }a^2+b-ab-a$
$\displaystyle \text{Answer:}$
$\displaystyle a^2+b-ab-a=a^2-ab-a+b$
$\displaystyle =a(a-b)-1(a-b)$
$\displaystyle =(a-b)(a-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Factorise using the grouping method: }(ax+by)^2+(bx-ay)^2$
$\displaystyle \text{Answer:}$
$\displaystyle (ax+by)^2+(bx-ay)^2$
$\displaystyle =a^2x^2+2abxy+b^2y^2+b^2x^2-2abxy+a^2y^2$
$\displaystyle =a^2x^2+b^2x^2+a^2y^2+b^2y^2$
$\displaystyle =x^2(a^2+b^2)+y^2(a^2+b^2)$
$\displaystyle =(a^2+b^2)(x^2+y^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Factorise using the grouping method: }a^2x^2+(ax^2+1)x+a$
$\displaystyle \text{Answer:}$
$\displaystyle a^2x^2+(ax^2+1)x+a=a^2x^2+ax^3+x+a$
$\displaystyle =ax^2(a+x)+1(x+a)$
$\displaystyle =(a+x)(ax^2+1)$

$\displaystyle \textbf{Question 14: }\text{Factorise using the grouping method: }a(a-4)-a+4$
$\displaystyle \text{Answer:}$
$\displaystyle a(a-4)-a+4=a(a-4)-1(a-4)$
$\displaystyle =(a-4)(a-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Factorise using the grouping method: }y^2-(a+b)y+ab$
$\displaystyle \text{Answer:}$
$\displaystyle y^2-(a+b)y+ab=y^2-ay-by+ab$
$\displaystyle =y(y-a)-b(y-a)$
$\displaystyle =(y-a)(y-b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 16: }\text{Factorise using the grouping method: }a^2+\frac{1}{a^2}-2-3a+\frac{3}{a}$
$\displaystyle \text{Answer:}$
$\displaystyle a^2+\frac{1}{a^2}-2-3a+\frac{3}{a}=\left(a-\frac{1}{a}\right)^2-3\left(a-\frac{1}{a}\right)$
$\displaystyle =\left(a-\frac{1}{a}\right)\left(a-\frac{1}{a}-3\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 17: }\text{Factorise using the grouping method: }x^2+y^2+x+y+2xy$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+y^2+x+y+2xy=(x^2+2xy+y^2)+(x+y)$
$\displaystyle =(x+y)^2+(x+y)$
$\displaystyle =(x+y)(x+y+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 18: }\text{Factorise using the grouping method: }a^2+4b^2-3a+6b-4ab$
$\displaystyle \text{Answer:}$
$\displaystyle a^2+4b^2-3a+6b-4ab=(a^2-4ab+4b^2)-3(a-2b)$
$\displaystyle =(a-2b)^2-3(a-2b)$
$\displaystyle =(a-2b)(a-2b-3)$
$\displaystyle \\$

$\displaystyle \textbf{Question 19: }\text{Factorise using the grouping method: }m(x-3y)^2+n(3y-x)+5x-15y$
$\displaystyle \text{Answer:}$
$\displaystyle n(3y-x)=-n(x-3y)\text{ and }5x-15y=5(x-3y)$
$\displaystyle \therefore m(x-3y)^2+n(3y-x)+5x-15y$
$\displaystyle =m(x-3y)^2-n(x-3y)+5(x-3y)$
$\displaystyle =(x-3y)\left[m(x-3y)-n+5\right]$
$\displaystyle \\$

$\displaystyle \textbf{Question 20: }\text{Factorise using the grouping method: }x(6x-5y)-4(6x-5y)^2$
$\displaystyle \text{Answer:}$
$\displaystyle x(6x-5y)-4(6x-5y)^2=(6x-5y)\left[x-4(6x-5y)\right]$
$\displaystyle =(6x-5y)(x-24x+20y)$
$\displaystyle =(6x-5y)(20y-23x)$

$\displaystyle \textbf{Exercise 5(B)}$

$\displaystyle \textbf{Question 1: }\text{Factorise: }a^2+10a+24$
$\displaystyle \text{Answer:}$
$\displaystyle a^2+10a+24=a^2+6a+4a+24$
$\displaystyle =a(a+6)+4(a+6)$
$\displaystyle =(a+6)(a+4)$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Factorise: }a^2-3a-40$
$\displaystyle \text{Answer:}$
$\displaystyle a^2-3a-40=a^2+5a-8a-40$
$\displaystyle =a(a+5)-8(a+5)$
$\displaystyle =(a+5)(a-8)$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Factorise: }1-2a-3a^2$
$\displaystyle \text{Answer:}$
$\displaystyle 1-2a-3a^2=1+a-3a-3a^2$
$\displaystyle =1(1+a)-3a(1+a)$
$\displaystyle =(1+a)(1-3a)$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Factorise: }x^2-3ax-88a^2$
$\displaystyle \text{Answer:}$
$\displaystyle x^2-3ax-88a^2=x^2+8ax-11ax-88a^2$
$\displaystyle =x(x+8a)-11a(x+8a)$
$\displaystyle =(x+8a)(x-11a)$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Factorise: }6a^2-a-15$
$\displaystyle \text{Answer:}$
$\displaystyle 6a^2-a-15=6a^2+9a-10a-15$
$\displaystyle =3a(2a+3)-5(2a+3)$
$\displaystyle =(2a+3)(3a-5)$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Factorise: }24a^3+37a^2-5a$
$\displaystyle \text{Answer:}$
$\displaystyle 24a^3+37a^2-5a=a(24a^2+37a-5)$
$\displaystyle =a(24a^2+40a-3a-5)$
$\displaystyle =8a^2(3a+5)-1(3a+5)$
$\displaystyle =a(3a+5)(8a-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{Factorise: }a(3a-2)-1$
$\displaystyle \text{Answer:}$
$\displaystyle a(3a-2)-1=3a^2-2a-1$
$\displaystyle =3a^2+a-3a-1$
$\displaystyle =a(3a+1)-1(3a+1)$
$\displaystyle =(a-1)(3a+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Factorise: }a^2b^2+8ab-9$
$\displaystyle \text{Answer:}$
$\displaystyle a^2b^2+8ab-9=(ab)^2+8(ab)-9$
$\displaystyle =(ab+9)(ab-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Factorise: }3-a(4+7a)$
$\displaystyle \text{Answer:}$
$\displaystyle 3-a(4+7a)=3-4a-7a^2$
$\displaystyle =3-7a+3a-7a^2$
$\displaystyle =1(3-7a)+a(3-7a)$
$\displaystyle =(a+1)(3-7a)$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Factorise: }(2a+b)^2-6a-3b-4$
$\displaystyle \text{Answer:}$
$\displaystyle (2a+b)^2-6a-3b-4=(2a+b)^2-3(2a+b)-4$
$\displaystyle =(2a+b)^2+(2a+b)-4(2a+b)-4$
$\displaystyle =(2a+b)(2a+b+1)-4(2a+b+1)$
$\displaystyle =(2a+b-4)(2a+b+1)$

$\displaystyle \textbf{Question 11: }\text{Factorise: }1-2a-2b-3(a+b)^2$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=a+b.$
$\displaystyle \therefore 1-2a-2b-3(a+b)^2=1-2x-3x^2$
$\displaystyle =1+x-3x-3x^2$
$\displaystyle =1(1+x)-3x(1+x)$
$\displaystyle =(1+x)(1-3x)$
$\displaystyle \therefore 1-2a-2b-3(a+b)^2=(1+a+b)(1-3a-3b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Factorise: }3a^2-1-2a$
$\displaystyle \text{Answer:}$
$\displaystyle 3a^2-1-2a=3a^2-2a-1$
$\displaystyle =3a^2-3a+a-1$
$\displaystyle =3a(a-1)+1(a-1)$
$\displaystyle =(a-1)(3a+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 13: }\text{Factorise: }x^2+3x+2+ax+2a$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+3x+2+ax+2a=(x^2+3x+2)+a(x+2)$
$\displaystyle =(x+1)(x+2)+a(x+2)$
$\displaystyle =(x+2)(x+a+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 14: }\text{Factorise: }(3x-2y)^2+3(3x-2y)-10$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }a=3x-2y.$
$\displaystyle \therefore (3x-2y)^2+3(3x-2y)-10=a^2+3a-10$
$\displaystyle =a^2+5a-2a-10$
$\displaystyle =a(a+5)-2(a+5)$
$\displaystyle =(a+5)(a-2)$
$\displaystyle \therefore (3x-2y)^2+3(3x-2y)-10=(3x-2y+5)(3x-2y-2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Factorise: }5-(3a^2-2a)(6-3a^2+2a)$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=3a^2-2a.$
$\displaystyle \therefore 5-(3a^2-2a)(6-3a^2+2a)=5-x(6-x)$
$\displaystyle =x^2-6x+5$
$\displaystyle =x^2-x-5x+5$
$\displaystyle =x(x-1)-5(x-1)$
$\displaystyle =(x-1)(x-5)$
$\displaystyle =(3a^2-2a-1)(3a^2-2a-5)$
$\displaystyle =(3a+1)(a-1)(3a-5)(a+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 16: }\text{For each trinomial, find whether it is factorisable or not. Factorise, if possible.}$
$\displaystyle \text{Answer:}$

$\displaystyle \text{(i) }x^2-3x-54$
$\displaystyle \text{Here, }a=1,\ b=-3,\ c=-54.$
$\displaystyle b^2-4ac=(-3)^2-4(1)(-54)=9+216=225=15^2$
$\displaystyle \therefore x^2-3x-54\text{ is factorisable.}$
$\displaystyle x^2-3x-54=x^2-9x+6x-54$
$\displaystyle =x(x-9)+6(x-9)$
$\displaystyle =(x-9)(x+6)$

$\displaystyle \text{(ii) }2x^2-7x-15$
$\displaystyle \text{Here, }a=2,\ b=-7,\ c=-15.$
$\displaystyle b^2-4ac=(-7)^2-4(2)(-15)=49+120=169=13^2$
$\displaystyle \therefore 2x^2-7x-15\text{ is factorisable.}$
$\displaystyle 2x^2-7x-15=2x^2-10x+3x-15$
$\displaystyle =2x(x-5)+3(x-5)$
$\displaystyle =(x-5)(2x+3)$

$\displaystyle \text{(iii) }2x^2+2x-75$
$\displaystyle \text{Here, }a=2,\ b=2,\ c=-75.$
$\displaystyle b^2-4ac=2^2-4(2)(-75)=4+600=604$
$\displaystyle \text{Since }604\text{ is not a perfect square, }2x^2+2x-75\text{ is not factorisable.}$

$\displaystyle \text{(iv) }3x^2+4x-10$
$\displaystyle \text{Here, }a=3,\ b=4,\ c=-10.$
$\displaystyle b^2-4ac=4^2-4(3)(-10)=16+120=136$
$\displaystyle \text{Since }136\text{ is not a perfect square, }3x^2+4x-10\text{ is not factorisable.}$

$\displaystyle \text{(v) }x(2x-1)-1$
$\displaystyle x(2x-1)-1=2x^2-x-1$
$\displaystyle \text{Here, }a=2,\ b=-1,\ c=-1.$
$\displaystyle b^2-4ac=(-1)^2-4(2)(-1)=1+8=9=3^2$
$\displaystyle \therefore 2x^2-x-1\text{ is factorisable.}$
$\displaystyle 2x^2-x-1=2x^2-2x+x-1$
$\displaystyle =2x(x-1)+1(x-1)$
$\displaystyle =(x-1)(2x+1)$

$\displaystyle \textbf{Exercise 5(C)}$

$\displaystyle \textbf{Question 1: }\text{Factorise: }25a^2-9b^2$
$\displaystyle \text{Answer:}$
$\displaystyle 25a^2-9b^2=(5a)^2-(3b)^2$
$\displaystyle =(5a+3b)(5a-3b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Factorise: }a^2-(2a+3b)^2$
$\displaystyle \text{Answer:}$
$\displaystyle a^2-(2a+3b)^2=\left[a+(2a+3b)\right]\left[a-(2a+3b)\right]$
$\displaystyle =(3a+3b)(-a-3b)$
$\displaystyle =-3(a+b)(a+3b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Factorise: }a^2-81(b-c)^2$
$\displaystyle \text{Answer:}$
$\displaystyle a^2-\left[9(b-c)\right]^2$
$\displaystyle =[a+9(b-c)][a-9(b-c)]$
$\displaystyle =(a+9b-9c)(a-9b+9c)$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Factorise: }25(2a-b)^2-81b^2$
$\displaystyle \text{Answer:}$
$\displaystyle [5(2a-b)]^2-(9b)^2$
$\displaystyle =[5(2a-b)+9b][5(2a-b)-9b]$
$\displaystyle =(10a+4b)(10a-14b)$
$\displaystyle =4(5a+2b)(5a-7b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Factorise: }50a^3-2a$
$\displaystyle \text{Answer:}$
$\displaystyle 50a^3-2a=2a(25a^2-1)$
$\displaystyle =2a\left[(5a)^2-1^2\right]$
$\displaystyle =2a(5a+1)(5a-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Factorise: }4a^2b-9b^3$
$\displaystyle \text{Answer:}$
$\displaystyle 4a^2b-9b^3=b(4a^2-9b^2)$
$\displaystyle =b\left[(2a)^2-(3b)^2\right]$
$\displaystyle =b(2a+3b)(2a-3b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{Factorise: }3a^5-108a^3$
$\displaystyle \text{Answer:}$
$\displaystyle 3a^5-108a^3=3a^3(a^2-36)$
$\displaystyle =3a^3(a+6)(a-6)$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Factorise: }9(a-2)^2-16(a+2)^2$
$\displaystyle \text{Answer:}$
$\displaystyle [3(a-2)]^2-[4(a+2)]^2$
$\displaystyle =[3(a-2)+4(a+2)][3(a-2)-4(a+2)]$
$\displaystyle =(7a+2)(-a-14)$
$\displaystyle =-(7a+2)(a+14)$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Factorise: }a^4-1$
$\displaystyle \text{Answer:}$
$\displaystyle a^4-1=(a^2)^2-1^2$
$\displaystyle =(a^2+1)(a^2-1)$
$\displaystyle =(a^2+1)(a+1)(a-1)$

$\displaystyle \textbf{Question 10: }\text{Factorise: }a^3+2a^2-a-2$
$\displaystyle \text{Answer:}$
$\displaystyle a^3+2a^2-a-2=a^2(a+2)-1(a+2)$
$\displaystyle =(a+2)(a^2-1)$
$\displaystyle =(a+2)(a+1)(a-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Factorise: }(a+b)^3-a-b$
$\displaystyle \text{Answer:}$
$\displaystyle (a+b)^3-a-b=(a+b)^3-(a+b)$
$\displaystyle =(a+b)\left[(a+b)^2-1\right]$
$\displaystyle =(a+b)(a+b+1)(a+b-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Factorise: }a(a-1)-b(b-1)$
$\displaystyle \text{Answer:}$
$\displaystyle a(a-1)-b(b-1)=a^2-a-b^2+b$
$\displaystyle =(a^2-b^2)-(a-b)$
$\displaystyle =(a-b)(a+b)-(a-b)$
$\displaystyle =(a-b)(a+b-1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 13: }\text{Factorise: }4a^2-(4b^2+4bc+c^2)$
$\displaystyle \text{Answer:}$
$\displaystyle 4a^2-(4b^2+4bc+c^2)=(2a)^2-(2b+c)^2$
$\displaystyle =[2a+(2b+c)][2a-(2b+c)]$
$\displaystyle =(2a+2b+c)(2a-2b-c)$
$\displaystyle \\$

$\displaystyle \textbf{Question 14: }\text{Factorise: }4a^2-49b^2+2a-7b$
$\displaystyle \text{Answer:}$
$\displaystyle 4a^2-49b^2+2a-7b=(2a-7b)(2a+7b)+(2a-7b)$
$\displaystyle =(2a-7b)(2a+7b+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Factorise: }9a^2+3a-8b-64b^2$
$\displaystyle \text{Answer:}$
$\displaystyle 9a^2+3a-8b-64b^2=(9a^2-64b^2)+(3a-8b)$
$\displaystyle =(3a-8b)(3a+8b)+(3a-8b)$
$\displaystyle =(3a-8b)(3a+8b+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 16: }\text{Factorise: }4a^2-12a+9-49b^2$
$\displaystyle \text{Answer:}$
$\displaystyle 4a^2-12a+9-49b^2=(2a-3)^2-(7b)^2$
$\displaystyle =(2a-3+7b)(2a-3-7b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 17: }\text{Factorise: }4xy-x^2-4y^2+z^2$
$\displaystyle \text{Answer:}$
$\displaystyle 4xy-x^2-4y^2+z^2=z^2-(x^2-4xy+4y^2)$
$\displaystyle =z^2-(x-2y)^2$
$\displaystyle =[z+(x-2y)][z-(x-2y)]$
$\displaystyle =(z+x-2y)(z-x+2y)$
$\displaystyle \\$

$\displaystyle \textbf{Question 18: }\text{Factorise: }a^2+b^2-c^2-d^2+2ab-2cd$
$\displaystyle \text{Answer:}$
$\displaystyle a^2+b^2-c^2-d^2+2ab-2cd=(a^2+2ab+b^2)-(c^2+2cd+d^2)$
$\displaystyle =(a+b)^2-(c+d)^2$
$\displaystyle =(a+b+c+d)(a+b-c-d)$
$\displaystyle \\$

$\displaystyle \textbf{Question 19: }\text{Factorise: }4x^2-12ax-y^2-z^2-2yz+9a^2$
$\displaystyle \text{Answer:}$
$\displaystyle 4x^2-12ax-y^2-z^2-2yz+9a^2=(4x^2-12ax+9a^2)-(y^2+2yz+z^2)$
$\displaystyle =(2x-3a)^2-(y+z)^2$
$\displaystyle =(2x-3a+y+z)(2x-3a-y-z)$

$\displaystyle \textbf{Question 20: }\text{Factorise: }(a^2-1)(b^2-1)+4ab$
$\displaystyle \text{Answer:}$
$\displaystyle (a^2-1)(b^2-1)+4ab=a^2b^2-a^2-b^2+1+4ab$
$\displaystyle =(ab+1)^2-(a-b)^2$
$\displaystyle =[(ab+1)+(a-b)][(ab+1)-(a-b)]$
$\displaystyle =(ab+a-b+1)(ab-a+b+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 21: }\text{Factorise: }x^4+x^2+1$
$\displaystyle \text{Answer:}$
$\displaystyle x^4+x^2+1=x^4+2x^2+1-x^2$
$\displaystyle =(x^2+1)^2-x^2$
$\displaystyle =(x^2+x+1)(x^2-x+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 22: }\text{Factorise: }(a^2+b^2-4c^2)^2-4a^2b^2$
$\displaystyle \text{Answer:}$
$\displaystyle (a^2+b^2-4c^2)^2-4a^2b^2=(a^2+b^2-4c^2)^2-(2ab)^2$
$\displaystyle =(a^2+b^2-4c^2+2ab)(a^2+b^2-4c^2-2ab)$
$\displaystyle =[(a+b)^2-(2c)^2][(a-b)^2-(2c)^2]$
$\displaystyle =(a+b+2c)(a+b-2c)(a-b+2c)(a-b-2c)$
$\displaystyle \\$

$\displaystyle \textbf{Question 23: }\text{Factorise: }(x^2+4y^2-9x^2y^2)^2-16x^2y^2$
$\displaystyle \text{Answer:}$
$\displaystyle (x^2+4y^2-9x^2y^2)^2-16x^2y^2$
$\displaystyle =(x^2+4y^2-9x^2y^2)^2-(4xy)^2$
$\displaystyle =(x^2+4y^2-9x^2y^2+4xy)(x^2+4y^2-9x^2y^2-4xy)$
$\displaystyle =[(x+2y)^2-(3xy)^2][(x-2y)^2-(3xy)^2]$
$\displaystyle =(x+2y+3xy)(x+2y-3xy)(x-2y+3xy)(x-2y-3xy)$
$\displaystyle \\$

$\displaystyle \textbf{Question 24: }\text{Factorise: }(a+b)^2-a^2+b^2$
$\displaystyle \text{Answer:}$
$\displaystyle (a+b)^2-a^2+b^2=a^2+2ab+b^2-a^2+b^2$
$\displaystyle =2ab+2b^2$
$\displaystyle =2b(a+b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 25: }\text{Factorise: }a^2-b^2-(a+b)^2$
$\displaystyle \text{Answer:}$
$\displaystyle a^2-b^2-(a+b)^2=(a+b)(a-b)-(a+b)^2$
$\displaystyle =(a+b)[(a-b)-(a+b)]$
$\displaystyle =(a+b)(-2b)$
$\displaystyle =-2b(a+b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 26: }\text{Factorise: }9a^2-(a^2-4)^2$
$\displaystyle \text{Answer:}$
$\displaystyle 9a^2-(a^2-4)^2=(3a)^2-(a^2-4)^2$
$\displaystyle =[3a+(a^2-4)][3a-(a^2-4)]$
$\displaystyle =(a^2+3a-4)(-a^2+3a+4)$
$\displaystyle =(a+4)(a-1)(4-a)(a+1)$
$\displaystyle =(a+4)(a-1)(a+1)(4-a)$
$\displaystyle \\$

$\displaystyle \textbf{Question 27: }\text{Factorise: }x^2+\frac{1}{x^2}-11$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+\frac{1}{x^2}-11=x^2+\frac{1}{x^2}-2-9$
$\displaystyle =\left(x-\frac{1}{x}\right)^2-3^2$
$\displaystyle =\left(x-\frac{1}{x}+3\right)\left(x-\frac{1}{x}-3\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 28: }\text{Factorise: }4x^2+\frac{1}{4x^2}+1$
$\displaystyle \text{Answer:}$
$\displaystyle 4x^2+\frac{1}{4x^2}+1=4x^2+\frac{1}{4x^2}+2-1$
$\displaystyle =\left(2x+\frac{1}{2x}\right)^2-1^2$
$\displaystyle =\left(2x+\frac{1}{2x}+1\right)\left(2x+\frac{1}{2x}-1\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 29: }\text{Factorise: }4a^4-a^2-12a-36$
$\displaystyle \text{Answer:}$
$\displaystyle 4a^4-a^2-12a-36=4a^4-(a^2+12a+36)$
$\displaystyle =(2a^2)^2-(a+6)^2$
$\displaystyle =(2a^2+a+6)(2a^2-a-6)$
$\displaystyle =(2a^2+a+6)(2a+3)(a-2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 30: }\text{Factorise: }a^2(b+c)-(b+c)^3$
$\displaystyle \text{Answer:}$
$\displaystyle a^2(b+c)-(b+c)^3=(b+c)\left[a^2-(b+c)^2\right]$
$\displaystyle =(b+c)[a+(b+c)][a-(b+c)]$
$\displaystyle =(b+c)(a+b+c)(a-b-c)$

$\displaystyle \textbf{Exercise 5(D)}$

$\displaystyle \textbf{Question 1: }\text{Factorise: }a^3-27$
$\displaystyle \text{Answer:}$
$\displaystyle a^3-27=a^3-3^3$
$\displaystyle =(a-3)(a^2+3a+9)$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Factorise: }1-8a^3$
$\displaystyle \text{Answer:}$
$\displaystyle 1-8a^3=1^3-(2a)^3$
$\displaystyle =(1-2a)(1+2a+4a^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Factorise: }64-a^3b^3$
$\displaystyle \text{Answer:}$
$\displaystyle 64-(ab)^3=4^3-(ab)^3$
$\displaystyle =(4-ab)(16+4ab+a^2b^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Factorise: }a^6+27b^3$
$\displaystyle \text{Answer:}$
$\displaystyle a^6+27b^3=(a^2)^3+(3b)^3$
$\displaystyle =(a^2+3b)(a^4-3a^2b+9b^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Factorise: }3x^7y-81x^4y^4$
$\displaystyle \text{Answer:}$
$\displaystyle 3x^7y-81x^4y^4=3x^4y(x^3-27y^3)$
$\displaystyle =3x^4y\left[x^3-(3y)^3\right]$
$\displaystyle =3x^4y(x-3y)(x^2+3xy+9y^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Factorise: }a^3-\frac{27}{a^3}$
$\displaystyle \text{Answer:}$
$\displaystyle a^3-\frac{27}{a^3}=a^3-\left(\frac{3}{a}\right)^3$
$\displaystyle =\left(a-\frac{3}{a}\right)\left(a^2+3+\frac{9}{a^2}\right)$

$\displaystyle \textbf{Question 7: }\text{Factorise: }a^3+0.064$
$\displaystyle \text{Answer:}$
$\displaystyle a^3+0.064=a^3+(0.4)^3$
$\displaystyle =(a+0.4)(a^2-0.4a+0.16)$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Factorise: }a^4-343a$
$\displaystyle \text{Answer:}$
$\displaystyle a^4-343a=a(a^3-343)$
$\displaystyle =a(a^3-7^3)$
$\displaystyle =a(a-7)(a^2+7a+49)$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Factorise: }(x-y)^3-8x^3$
$\displaystyle \text{Answer:}$
$\displaystyle (x-y)^3-8x^3=(x-y)^3-(2x)^3$
$\displaystyle =[(x-y)-2x]\left[(x-y)^2+2x(x-y)+4x^2\right]$
$\displaystyle =-(x+y)(7x^2-4xy+y^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Factorise: }\frac{8a^3}{27}-\frac{b^3}{8}$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{8a^3}{27}-\frac{b^3}{8}=\left(\frac{2a}{3}\right)^3-\left(\frac{b}{2}\right)^3$
$\displaystyle =\left(\frac{2a}{3}-\frac{b}{2}\right)\left(\frac{4a^2}{9}+\frac{ab}{3}+\frac{b^2}{4}\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Factorise: }a^6-b^6$
$\displaystyle \text{Answer:}$
$\displaystyle a^6-b^6=(a^3)^2-(b^3)^2$
$\displaystyle =(a^3-b^3)(a^3+b^3)$
$\displaystyle =(a-b)(a^2+ab+b^2)(a+b)(a^2-ab+b^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Factorise: }a^6-7a^3-8$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=a^3.$
$\displaystyle \therefore a^6-7a^3-8=x^2-7x-8$
$\displaystyle =x^2-8x+x-8$
$\displaystyle =x(x-8)+1(x-8)$
$\displaystyle =(x-8)(x+1)$
$\displaystyle =(a^3-8)(a^3+1)$
$\displaystyle =(a-2)(a^2+2a+4)(a+1)(a^2-a+1)$

$\displaystyle \textbf{Question 13: }\text{Factorise: }a^3-27b^3+2a^2b-6ab^2$
$\displaystyle \text{Answer:}$
$\displaystyle a^3-27b^3+2a^2b-6ab^2=\left[a^3-(3b)^3\right]+2ab(a-3b)$
$\displaystyle =(a-3b)(a^2+3ab+9b^2)+2ab(a-3b)$
$\displaystyle =(a-3b)(a^2+3ab+9b^2+2ab)$
$\displaystyle =(a-3b)(a^2+5ab+9b^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 14: }\text{Factorise: }8a^3-b^3-4ax+2bx$
$\displaystyle \text{Answer:}$
$\displaystyle 8a^3-b^3-4ax+2bx=\left[(2a)^3-b^3\right]-2x(2a-b)$
$\displaystyle =(2a-b)(4a^2+2ab+b^2)-2x(2a-b)$
$\displaystyle =(2a-b)(4a^2+2ab+b^2-2x)$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Factorise: }a-b-a^3+b^3$
$\displaystyle \text{Answer:}$
$\displaystyle a-b-a^3+b^3=(a-b)-(a^3-b^3)$
$\displaystyle =(a-b)-(a-b)(a^2+ab+b^2)$
$\displaystyle =(a-b)(1-a^2-ab-b^2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 16: }\text{Factorise: }2x^3+54y^3-4x-12y$
$\displaystyle \text{Answer:}$
$\displaystyle 2x^3+54y^3-4x-12y=2(x^3+27y^3)-4(x+3y)$
$\displaystyle =2\left[x^3+(3y)^3\right]-4(x+3y)$
$\displaystyle =2(x+3y)(x^2-3xy+9y^2)-4(x+3y)$
$\displaystyle =2(x+3y)(x^2-3xy+9y^2-2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 17: }\text{Show that:}$
$\displaystyle \text{(i) }13^3-5^3\text{ is divisible by }8.$
$\displaystyle \text{(ii) }35^3+27^3\text{ is divisible by }62.$
$\displaystyle \text{Answer:}$

$\displaystyle \text{(i) Using }a^3-b^3=(a-b)(a^2+ab+b^2),$
$\displaystyle 13^3-5^3=(13-5)(13^2+13\times5+5^2)$
$\displaystyle =8(169+65+25)$
$\displaystyle =8\times259$
$\displaystyle \therefore 13^3-5^3\text{ is divisible by }8.$

$\displaystyle \text{(ii) Using }a^3+b^3=(a+b)(a^2-ab+b^2),$
$\displaystyle 35^3+27^3=(35+27)(35^2-35\times27+27^2)$
$\displaystyle =62(1225-945+729)$
$\displaystyle =62\times1009$
$\displaystyle \therefore 35^3+27^3\text{ is divisible by }62.$

$\displaystyle \textbf{Exercise 5(E)}$

$\displaystyle \textbf{Question 1: }\text{Factorise: }x^2+\frac{1}{4x^2}+1-7x-\frac{7}{2x}$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+\frac{1}{4x^2}+1-7x-\frac{7}{2x}=\left(x+\frac{1}{2x}\right)^2-7\left(x+\frac{1}{2x}\right)$
$\displaystyle =\left(x+\frac{1}{2x}\right)\left(x+\frac{1}{2x}-7\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 2: }\text{Factorise: }9a^2+\frac{1}{9a^2}-2-12a+\frac{4}{3a}$
$\displaystyle \text{Answer:}$
$\displaystyle 9a^2+\frac{1}{9a^2}-2-12a+\frac{4}{3a}=\left(3a-\frac{1}{3a}\right)^2-4\left(3a-\frac{1}{3a}\right)$
$\displaystyle =\left(3a-\frac{1}{3a}\right)\left(3a-\frac{1}{3a}-4\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 3: }\text{Factorise: }x^2+\frac{a^2+1}{a}x+1$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+\frac{a^2+1}{a}x+1=x^2+\left(a+\frac{1}{a}\right)x+1$
$\displaystyle =x^2+ax+\frac{x}{a}+1$
$\displaystyle =x(x+a)+\frac{1}{a}(x+a)$
$\displaystyle =(x+a)\left(x+\frac{1}{a}\right)$
$\displaystyle \\$

$\displaystyle \textbf{Question 4: }\text{Factorise: }x^4+y^4-27x^2y^2$
$\displaystyle \text{Answer:}$
$\displaystyle x^4+y^4-27x^2y^2=x^4-2x^2y^2+y^4-25x^2y^2$
$\displaystyle =(x^2-y^2)^2-(5xy)^2$
$\displaystyle =(x^2-y^2+5xy)(x^2-y^2-5xy)$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Factorise: }4x^4+9y^4+11x^2y^2$
$\displaystyle \text{Answer:}$
$\displaystyle 4x^4+9y^4+11x^2y^2=4x^4+12x^2y^2+9y^4-x^2y^2$
$\displaystyle =(2x^2+3y^2)^2-(xy)^2$
$\displaystyle =(2x^2+3y^2+xy)(2x^2+3y^2-xy)$
$\displaystyle \\$

$\displaystyle \textbf{Question 6: }\text{Factorise: }x^2+\frac{1}{x^2}-3$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+\frac{1}{x^2}-3=x^2+\frac{1}{x^2}-2-1$
$\displaystyle =\left(x-\frac{1}{x}\right)^2-1^2$
$\displaystyle =\left(x-\frac{1}{x}+1\right)\left(x-\frac{1}{x}-1\right)$

$\displaystyle \textbf{Question 7: }\text{Factorise: }a-b-4a^2+4b^2$
$\displaystyle \text{Answer:}$
$\displaystyle a-b-4a^2+4b^2=(a-b)-4(a^2-b^2)$
$\displaystyle =(a-b)-4(a-b)(a+b)$
$\displaystyle =(a-b)\left[1-4(a+b)\right]$
$\displaystyle =(a-b)(1-4a-4b)$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Factorise: }(2a-3)^2-2(2a-3)(a-1)+(a-1)^2$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Using }x^2-2xy+y^2=(x-y)^2,$
$\displaystyle (2a-3)^2-2(2a-3)(a-1)+(a-1)^2$
$\displaystyle =\left[(2a-3)-(a-1)\right]^2$
$\displaystyle =(a-2)^2$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Factorise: }(a^2-3a)(a^2-3a+7)+10$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=a^2-3a.$
$\displaystyle \therefore (a^2-3a)(a^2-3a+7)+10=x(x+7)+10$
$\displaystyle =x^2+7x+10$
$\displaystyle =(x+5)(x+2)$
$\displaystyle =(a^2-3a+5)(a^2-3a+2)$
$\displaystyle =(a^2-3a+5)(a-1)(a-2)$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Factorise: }(a^2-a)(4a^2-4a-5)-6$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }x=a^2-a.$
$\displaystyle \therefore (a^2-a)(4a^2-4a-5)-6=x(4x-5)-6$
$\displaystyle =4x^2-5x-6$
$\displaystyle =4x^2-8x+3x-6$
$\displaystyle =4x(x-2)+3(x-2)$
$\displaystyle =(x-2)(4x+3)$
$\displaystyle =(a^2-a-2)(4a^2-4a+3)$
$\displaystyle =(a-2)(a+1)(4a^2-4a+3)$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Factorise: }x^4+y^4-3x^2y^2$
$\displaystyle \text{Answer:}$
$\displaystyle x^4+y^4-3x^2y^2=x^4-2x^2y^2+y^4-x^2y^2$
$\displaystyle =(x^2-y^2)^2-(xy)^2$
$\displaystyle =(x^2-y^2+xy)(x^2-y^2-xy)$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{Factorise: }5a^2-b^2-4ab+7a-7b$
$\displaystyle \text{Answer:}$
$\displaystyle 5a^2-b^2-4ab+7a-7b=(5a^2-5ab)+(ab-b^2)+7(a-b)$
$\displaystyle =5a(a-b)+b(a-b)+7(a-b)$
$\displaystyle =(a-b)(5a+b+7)$
$\displaystyle \\$

$\displaystyle \textbf{Question 13: }\text{Factorise: }12(3x-2y)^2-3x+2y-1$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }a=3x-2y.$
$\displaystyle \therefore 12(3x-2y)^2-3x+2y-1=12a^2-a-1$
$\displaystyle =12a^2-4a+3a-1$
$\displaystyle =4a(3a-1)+1(3a-1)$
$\displaystyle =(3a-1)(4a+1)$
$\displaystyle =[3(3x-2y)-1][4(3x-2y)+1]$
$\displaystyle =(9x-6y-1)(12x-8y+1)$

$\displaystyle \textbf{Question 14: }\text{Factorise: }4(2x-3y)^2-8x+12y-3$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }a=2x-3y.$
$\displaystyle \therefore 4(2x-3y)^2-8x+12y-3=4a^2-4a-3$
$\displaystyle =4a^2+2a-6a-3$
$\displaystyle =2a(2a+1)-3(2a+1)$
$\displaystyle =(2a+1)(2a-3)$
$\displaystyle =[2(2x-3y)+1][2(2x-3y)-3]$
$\displaystyle =(4x-6y+1)(4x-6y-3)$
$\displaystyle \\$

$\displaystyle \textbf{Question 15: }\text{Factorise: }3-5x+5y-12(x-y)^2$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }a=x-y.$
$\displaystyle \therefore 3-5x+5y-12(x-y)^2=3-5a-12a^2$
$\displaystyle =3+4a-9a-12a^2$
$\displaystyle =1(3+4a)-3a(3+4a)$
$\displaystyle =(1-3a)(4a+3)$
$\displaystyle =[1-3(x-y)][4(x-y)+3]$
$\displaystyle =(1-3x+3y)(4x-4y+3)$
$\displaystyle \\$

$\displaystyle \textbf{Question 16: }\text{Factorise: }9x^2+3x-8y-64y^2$
$\displaystyle \text{Answer:}$
$\displaystyle 9x^2+3x-8y-64y^2=(9x^2-64y^2)+(3x-8y)$
$\displaystyle =(3x-8y)(3x+8y)+(3x-8y)$
$\displaystyle =(3x-8y)(3x+8y+1)$
$\displaystyle \\$

$\displaystyle \textbf{Question 17: }\text{Factorise: }2\sqrt3x^2+x-5\sqrt3$
$\displaystyle \text{Answer:}$
$\displaystyle 2\sqrt3x^2+x-5\sqrt3=2\sqrt3x^2+6x-5x-5\sqrt3$
$\displaystyle =2x(\sqrt3x+3)-5(x+\sqrt3)$
$\displaystyle =2\sqrt3x(x+\sqrt3)-5(x+\sqrt3)$
$\displaystyle =(x+\sqrt3)(2\sqrt3x-5)$
$\displaystyle \\$

$\displaystyle \textbf{Question 18: }\text{Factorise: }\frac14(a+b)^2-\frac9{16}(2a-b)^2$
$\displaystyle \text{Answer:}$
$\displaystyle \frac14(a+b)^2-\frac9{16}(2a-b)^2=\left(\frac{a+b}{2}\right)^2-\left[\frac{3(2a-b)}{4}\right]^2$
$\displaystyle =\left[\frac{a+b}{2}+\frac{3(2a-b)}{4}\right]\left[\frac{a+b}{2}-\frac{3(2a-b)}{4}\right]$
$\displaystyle =\frac14(8a-b)\times\frac14(5b-4a)$
$\displaystyle =\frac1{16}(8a-b)(5b-4a)$
$\displaystyle \\$

$\displaystyle \textbf{Question 19: }\text{Factorise: }2(ab+cd)-a^2-b^2+c^2+d^2$
$\displaystyle \text{Answer:}$
$\displaystyle 2(ab+cd)-a^2-b^2+c^2+d^2$
$\displaystyle =c^2+2cd+d^2-(a^2-2ab+b^2)$
$\displaystyle =(c+d)^2-(a-b)^2$
$\displaystyle =[(c+d)+(a-b)][(c+d)-(a-b)]$
$\displaystyle =(a-b+c+d)(-a+b+c+d)$

$\displaystyle \textbf{Question 20: }\text{Find the value of:}$
$\displaystyle \text{(i) }(987)^2-(13)^2$
$\displaystyle \text{Answer:}$
$\displaystyle (987)^2-(13)^2=(987-13)(987+13)$
$\displaystyle =974\times1000$
$\displaystyle =974000$
$\displaystyle \\$

$\displaystyle \text{(ii) }(67.8)^2-(32.2)^2$
$\displaystyle \text{Answer:}$
$\displaystyle (67.8)^2-(32.2)^2=(67.8-32.2)(67.8+32.2)$
$\displaystyle =35.6\times100$
$\displaystyle =3560$
$\displaystyle \\$

$\displaystyle \text{(iii) }\frac{(6.7)^2-(3.3)^2}{6.7-3.3}$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{(6.7)^2-(3.3)^2}{6.7-3.3}=\frac{(6.7-3.3)(6.7+3.3)}{6.7-3.3}$
$\displaystyle =6.7+3.3$
$\displaystyle =10$
$\displaystyle \\$

$\displaystyle \text{(iv) }\frac{(18.5)^2-(6.5)^2}{18.5+6.5}$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{(18.5)^2-(6.5)^2}{18.5+6.5}=\frac{(18.5-6.5)(18.5+6.5)}{18.5+6.5}$
$\displaystyle =18.5-6.5$
$\displaystyle =12$

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Ultimate Math Study Guide for Students

Class 9: Factorization – Exercise 5

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