RD Sharma solution class 8 chapter 7 Factorization Exercise 7.5

Exercise 7.5

Page-7.17

Question 1:

Factorize each of the following expression:
16x² − 25y²

Answer 1:

$$\begin{gathered} 16x^2-25y^2 \\ =(4x{)}^2-(5y{)}^2 \\ =(4x-5y)(4x+5y) \end{gathered}$$

Question 2:

Factorize each of the following expression:
27x² − 12y²

Answer 2:

$$\begin{gathered} 27x^2-12y^2 \\ =3(9x^2-4y^2) \\ =3\left[\right(3x{)}^2-\left(2y{)}^2\right] \\ =3(3x-2y)(3x+2y) \end{gathered}$$

Question 3:

Factorize each of the following expression:
144a² − 289b²

Answer 3:

$$\begin{gathered} 144a^2-289b^2 \\ =(12a{)}^2-(17b{)}^2 \\ =(12a-17b)(12a+17b) \end{gathered}$$

Question 4:

Factorize each of the following expression:
12m² − 27

Answer 4:

$$\begin{gathered} 12m^2-27 \\ =3(4m^2-9) \\ =3\left[\right(2m{)}^2-3^2] \\ =3(2m-3\left)\right(2m+3) \end{gathered}$$

Question 5:

Factorize each of the following expression:
125x² − 45y²

Answer 5:

$$\begin{gathered} 125x^2-45y^2 \\ =5(25x^2-9y^2) \\ =5\left[\right(5x{)}^2-\left(3y{)}^2\right] \\ =5(5x-3y)(5x+3y) \end{gathered}$$

Question 6:

Factorize each of the following expression:
144a² − 169b²

Answer 6:

$$\begin{gathered} 144a^2-169b^2 \\ =(12a{)}^2-(13b{)}^2 \\ =(12a-13b)(12a+13b) \end{gathered}$$

Question 7:

Factorize each of the following expression:
(2a − b)² − 16c²

Answer 7:

$$\begin{gathered} (2a-b{)}^2-16c^2 \\ =(2a-b{)}^2-(4c{)}^2 \\ =[(2a-b)-4c\left]\right[(2a-b)+4c] \\ =(2a-b-4c\left)\right(2a-b+4c) \end{gathered}$$

Question 8:

Factorize each of the following expression:
(x + 2y)² − 4(2x − y)²

Answer 8:

$$\begin{gathered} (x+2y{)}^2-4(2x-y{)}^2=(x+2y{)}^2-[2(2x-y){]}^2 \\ =\left[\right(x+2y)-2(2x-y\left)\right]\left[\right(x+2y)+2(2x-y\left)\right] \\ =(x+2y-4x+2y)(x+2y+4x-2y) \\ =5x(4y-3x) \end{gathered}$$

Question 9:

Factorize each of the following expression:
3a⁵ − 48a³

Answer 9:

$$\begin{gathered} 3a^5-48a^3 \\ =3a^3(a^2-16) \\ =3a^3(a^2-4^2) \\ =3a^3(a-4)(a+4) \end{gathered}$$

Question 10:

Factorize each of the following expression:
a⁴ − 16b⁴

Answer 10:

$$\begin{gathered} a^4-16b^4=a^4-2^4{b}^4=(a^2{)}^2-(2^2{b}^2{)}^2 \\ =(a^2-2^2{b}^2)(a^2+2^2{b}^2) \\ =[a^2-(2b{)}^2\left]\right(a^2+4b^2) \\ =(a-2b\left)\right(a+2b\left)\right(a^2+4b^2) \end{gathered}$$

Question 11:

Factorize each of the following expression:
x⁸ − 1

Answer 11:

$$\begin{gathered} x^8-1 \\ =(x^4{)}^2-1^2 \\ =(x^4-1\left)\right(x^4+1) \\ =[(x^2{)}^2-1^2](x^4+1) \\ =(x^2-1)(x^2+1)(x^4+1) \\ =(x^2-1^2)(x^2+1)(x^4+1) \\ =(x-1)(x+1)(x^2+1)(x^4+1) \end{gathered}$$

Question 12:

Factorize each of the following expression:
64 − (a + 1)²

Answer 12:

$$\begin{gathered} 64-(a+1{)}^2 \\ =(8{)}^2-(a+1{)}^2 \\ =[8-(a+1)\left]\right[8+(a+1)] \\ =(8-a-1\left)\right(8+a+1) \\ =(7-a\left)\right(9+a) \end{gathered}$$

Question 13:

Factorize each of the following expression:
36l² − (m + n)²

Answer 13:

$$\begin{gathered} 36l^2-(m+n{)}^2 \\ =(6l{)}^2-(m+n{)}^2 \\ =[6l-(m+n)\left]\right[6l+(m+n)] \\ =(6l-m-n\left)\right(6l+m+n) \end{gathered}$$

Question 14:

Factorize each of the following expression:
25x⁴y⁴ − 1

Answer 14:

$$\begin{gathered} 25x^4{y}^4-1 \\ =(5x^2{y}^2{)}^2-1 \\ =(5x^2{y}^2-1\left)\right(5x^2{y}^2+1) \end{gathered}$$

Question 15:

Factorize each of the following expression:
$a^4-\frac{1}{b^4}$

Answer 15:

$a$⁴ - 1/b⁴
= (a²)² - 1/(b²)²
= a²- 1/b²a² + 1/b²
= a - 1/ba + 1/ba² + 1/b²

Question 16:

Factorize each of the following expression:
x³ − 144x

Answer 16:

$$\begin{gathered} x^3-144x \\ =x(x^2-144) \\ =x(x^2-{12}^2) \\ =x(x-12)(x+12) \end{gathered}$$

Question 17:

Factorize each of the following expression:
(x - 4y)² − 625

Answer 17:

$$\begin{gathered} (x-4y{)}^2-625 \\ =(x-4y{)}^2-{25}^2 \\ =\left[\right(x-4y)-25]\left[\right(x-4y)+25] \\ =(x-4y-25)(x-4y+25) \end{gathered}$$

Question 18:

Factorize each of the following expression:
9(a − b)² − 100(x − y)²

Answer 18:

$$\begin{gathered} 9(a-b{)}^2-100(x-y{)}^2 \\ =\left[3\right(a-b){]}^2-[10(x-y){]}^2 \\ =\left[3\right(a-b)-10(x-y\left)\right]\left[3\right(a-b)+10(x-y\left)\right] \\ =(3a-3b-10x+10y)(3a-3b+10x-10y) \end{gathered}$$

Question 19:

Factorize each of the following expression:
(3 + 2a)² − 25a²

Answer 19:

$$\begin{gathered} (3+2a{)}^2-25a^2 \\ =(3+2a{)}^2-(5a{)}^2 \\ =[(3+2a)-5a\left]\right[(3+2a)+5a] \\ =(3+2a-5a\left)\right(3+2a+5a) \\ =(3-3a\left)\right(3+7a) \\ =3(1-a\left)\right(3+7a) \end{gathered}$$

Question 20:

Factorize each of the following expression:
(x + y)² − (a − b)²

Answer 20:

$$\begin{gathered} (x+y{)}^2-(a-b{)}^2 \\ =\left[\right(x+y)-(a-b\left)\right]\left[\right(x+y)+(a-b\left)\right] \\ =(x+y-a+b)(x+y+a-b) \end{gathered}$$

Question 21:

Factorize each of the following expression:
$\frac{1}{16}{x}^2{y}^2-\frac{4}{49}{y}^2{z}^2$

Answer 21:

$$\begin{gathered} \frac{1}{16}{x}^2{y}^2-\frac{4}{49}{y}^2{z}^2 \\ =y^2\left(\frac{1}{16}{x}^2-\frac{4}{49}{z}^2\right) \\ =y^2\left[{\left(\frac{1}{4}x\right)}^2-{\left(\frac{2}{7}z\right)}^2\right] \\ =y^2\left(\frac{1}{4}x-\frac{2}{7}z\right)\left(\frac{1}{4}x+\frac{2}{7}z\right) \\ =y^2\left(\frac{x}{4}-\frac{2}{7}z\right)\left(\frac{x}{4}+\frac{2}{7}z\right) \end{gathered}$$

Question 22:

Factorize each of the following expression:
75a³b² - 108ab⁴

Answer 22:

$$\begin{gathered} 75a^3{b}^2-108a{b}^4 \\ =3a{b}^2(25a^2-36b^2) \\ =3a{b}^2\left[\right(5a{)}^2-\left(6b{)}^2\right] \\ =3a{b}^2(5a-6b)(5a+6b) \end{gathered}$$

Question 23:

Factorize each of the following expression:
x⁵ − 16x³

Answer 23:

$$\begin{gathered} x^5-16x^3 \\ =x^3(x^2-16) \\ =x^3(x^2-4^2) \\ =x^3(x-4)(x+4) \end{gathered}$$

Question 24:

Factorize each of the following expression:
$\frac{50}{x^2}-\frac{2x^2}{81}$

Answer 24:

$$\begin{gathered} \frac{50}{{\mathrm{x}}^2}-\frac{2{\mathrm{x}}^2}{81} \\ =2\left(\frac{25}{{\mathrm{x}}^2}-\frac{{\mathrm{x}}^2}{81}\right) \\ =2\left\{{\left(\frac{5}{\mathrm{x}}\right)}^2-{\left(\frac{\mathrm{x}}{9}\right)}^2\right\} \\ =2\left(\frac{5}{\mathrm{x}}-\frac{\mathrm{x}}{9}\right)\left(\frac{5}{\mathrm{x}}+\frac{\mathrm{x}}{9}\right) \end{gathered}$$

Question 25:

Factorize each of the following expression:
256x⁵ − 81x

Answer 25:

$$\begin{gathered} 256x^5-81x \\ =x(256x^4-81) \\ =x\left[\right(16x^2{)}^2-9^2] \\ =x(16x^2+9\left)\right(16x^2-9) \\ =x(16x^2+9\left)\right[(4x{)}^2-3^2] \\ =x(16x^2+9)(4x+3)(4x-3) \end{gathered}$$

Question 26:

Factorize each of the following expression:
a⁴ − (2b + c)⁴

Answer 26:

$$\begin{gathered} a^4-(2b+c{)}^4 \\ =(a^2{)}^2-\left[\right(2b+c{)}^2{]}^2 \\ =[a^2+(2b+c{)}^2\left]\right[a^2-(2b+c{)}^2] \\ =[a^2+(2b+c{)}^2\left]\right\{[a+(2b+c\left)\right][a-(2b+c\left)\right]\} \\ =[a^2+(2b+c{)}^2](a+2b+c)(a-2b-c) \end{gathered}$$

Question 27:

Factorize each of the following expression:
(3x + 4y)⁴ − x⁴

Answer 27:

$$\begin{gathered} (3\mathrm{x}+4\mathrm{y}{)}^4-{\mathrm{x}}^4 \\ =[(3\mathrm{x}+4\mathrm{y}{)}^2{]}^2-({\mathrm{x}}^2{)}^2 \\ =\left[\right(3\mathrm{x}+4\mathrm{y}{)}^2+{\mathrm{x}}^2\left]\right[(3\mathrm{x}+4\mathrm{y}{)}^2-{\mathrm{x}}^2] \\ =\left[\right(3\mathrm{x}+4\mathrm{y}{)}^2+{\mathrm{x}}^2\left]\right[(3\mathrm{x}+4\mathrm{y})+\mathrm{x}\left]\right[(3\mathrm{x}+4\mathrm{y})-\mathrm{x}] \\ =\{(3\mathrm{x}+4\mathrm{y}{)}^2+{\mathrm{x}}^2\}(3\mathrm{x}+4\mathrm{y}+\mathrm{x})(3\mathrm{x}+4\mathrm{y}-\mathrm{x}) \\ =\left\{{\left(3\mathrm{x}+4\mathrm{y}\right)}^2+{\mathrm{x}}^2\right\}(4\mathrm{x}+4\mathrm{y})(2\mathrm{x}+4\mathrm{y}) \\ =\left\{{\left(3\mathrm{x}+4\mathrm{y}\right)}^2+{\mathrm{x}}^2\right\}4(\mathrm{x}+\mathrm{y})2(\mathrm{x}+2\mathrm{y}) \\ =8\left\{{\left(3\mathrm{x}+4\mathrm{y}\right)}^2+{\mathrm{x}}^2\right\}(\mathrm{x}+\mathrm{y})(\mathrm{x}+2\mathrm{y}) \end{gathered}$$

Question 28:

Factorize each of the following expression:
p²q² − p⁴q⁴

Answer 28:

$$\begin{gathered} p^2{q}^2-p^4{q}^4 \\ =p^2{q}^2(1-p^2{q}^2) \\ =p^2{q}^2[1-(pq{)}^2] \\ =p^2{q}^2(1-pq\left)\right(1+pq) \end{gathered}$$

Question 29:

Factorize each of the following expression:
3x³y − 243xy³

Answer 29:

$$\begin{gathered} 3x^{3}y-243x{y}^3 \\ =3xy(x^2-81y^2) \\ =3xy[x^2-(9y{)}^2] \\ =3xy(x-9y\left)\right(x+9y) \end{gathered}$$

Question 30:

Factorize each of the following expression:
a⁴b⁴ − 16c⁴

Answer 30:

$$\begin{gathered} a^4{b}^4-16c^4 \\ =\left[\right(a^2{b}^2{)}^2-\left(4c^2{)}^2\right] \\ =(a^2{b}^2+4c^2)(a^2{b}^2-4c^2) \\ =(a^2{b}^2+4c^2)\left[\right(ab{)}^2-\left(2c{)}^2\right] \\ =(a^2{b}^2+4c^2)(ab+2c)(ab-2c) \end{gathered}$$

Question 31:

Factorize each of the following expression:
x⁴ − 625

Answer 31:

$$\begin{gathered} x^4-625 \\ =(x^2{)}^2-{25}^2 \\ =(x^2+25\left)\right(x^2-25) \\ =(x^2+25\left)\right(x^2-5^2) \\ =(x^2+25\left)\right(x+5\left)\right(x-5) \end{gathered}$$

Question 32:

Factorize each of the following expression:
x⁴ − 1

Answer 32:

$$\begin{gathered} x^4-1 \\ =(x^2{)}^2-1 \\ =(x^2+1\left)\right(x^2-1) \\ =(x^2+1\left)\right(x+1\left)\right(x-1) \end{gathered}$$

Question 33:

Factorize each of the following expression:
49(a − b)² − 25(a + b)²

Answer 33:

$$\begin{gathered} 49(a-b{)}^2-25(a+b{)}^2 \\ =\left[7\right(a-b){]}^2-[5(a+b){]}^2 \\ =\left[7\right(a-b)-5(a+b\left)\right]\left[7\right(a-b)+5(a+b\left)\right] \\ =(7a-7b-5a-5b)(7a-7b+5a+5b) \\ =(2a-12b)(12a-2b) \\ =2(a-6b)2(6a-b) \\ =4(a-6b)(6a-b) \end{gathered}$$

Question 34:

Factorize each of the following expression:
x − y − x² + y²

Answer 34:

$$\begin{gathered} x-y-x^2+y^2 \\ =(x-y)+(y^2-x^2) [Regrouping the terms] \\ =(x-y)+(y+x)(y-x) \\ =(x-y)-(y+x)(x-y) [\because (y-x)=-(x-y\left)\right] \\ =(x-y)[1-(y+x\left)\right] \\ =(x-y)(1-x-y) \end{gathered}$$

Question 35:

Factorize each of the following expression:
16(2x − 1)² − 25y²

Answer 35:

$$\begin{gathered} 16(2x-1{)}^2-25y^2 \\ =[4(2x-1){]}^2-(5y{)}^2 \\ =[4(2x-1)-5y\left]\right[4(2x-1)+5y] \\ =(8x-4-5y\left)\right(8x-4+5y) \\ =(8x-5y-4\left)\right(8x+5y-4) \end{gathered}$$

Question 36:

Factorize each of the following expression:
4(xy + 1)² − 9(x − 1)²

Answer 36:

$$\begin{gathered} 4(xy+1{)}^2-9(x-1{)}^2 \\ =\left[2\right(xy+1){]}^2-[3(x-1){]}^2 \\ =\left[2\right(xy+1)-3(x-1\left)\right]\left[2\right(xy+1)+3(x-1\left)\right] \\ =(2xy+2-3x+3)(2xy+2+3x-3) \\ =(2xy-3x+5)(2xy+3x-1) \end{gathered}$$

Question 37:

Factorize each of the following expression:
(2x + 1)² − 9x⁴

Answer 37:

$$\begin{gathered} (2\mathrm{x}+1{)}^2-9{\mathrm{x}}^4 \\ =(2\mathrm{x}+1{)}^2-(3{\mathrm{x}}^2{)}^2 \\ =[(2\mathrm{x}+1)-3{\mathrm{x}}^2\left]\right[(2\mathrm{x}+1)+3{\mathrm{x}}^2] \\ =(-3{\mathrm{x}}^2+2\mathrm{x}+1\left)\right(3{\mathrm{x}}^2+2\mathrm{x}+1) \\ \mathrm{We} \mathrm{can} \mathrm{factorise} \mathrm{the} \mathrm{quadratic} \mathrm{expressions} \mathrm{in} \mathrm{the} \mathrm{curved} \mathrm{brackets} \mathrm{as}: \\ (-3{\mathrm{x}}^2+3\mathrm{x}-\mathrm{x}+1\left)\right(3{\mathrm{x}}^2+2\mathrm{x}+1) \\ =\left\{3\mathrm{x}(-\mathrm{x}+1)+1(-\mathrm{x}+1)\right\}(3{\mathrm{x}}^2+2\mathrm{x}+1) \\ =(-\mathrm{x}+1\left)\right(3\mathrm{x}+1\left)\right(3{\mathrm{x}}^2+2\mathrm{x}+1) \\ =-(\mathrm{x}-1\left)\right(3\mathrm{x}+1\left)\right(3{\mathrm{x}}^2+2\mathrm{x}+1) \end{gathered}$$

Question 38:

Factorize each of the following expression:
x⁴ − (2y − 3z)²

Answer 38:

$$\begin{gathered} {\mathrm{x}}^4-(2\mathrm{y}-3\mathrm{z}{)}^2 \\ =({\mathrm{x}}^2{)}^2-(2\mathrm{y}-3\mathrm{z}{)}^2 \\ =[{\mathrm{x}}^2-(2\mathrm{y}-3\mathrm{z})\left]\right[{\mathrm{x}}^2+(2\mathrm{y}-3\mathrm{z})] \\ =({\mathrm{x}}^2-2\mathrm{y}+3\mathrm{z}\left)\right({\mathrm{x}}^2+2\mathrm{y}-3\mathrm{z}) \end{gathered}$$

Question 39:

Factorize each of the following expression:
a² − b² + a − b

Answer 39:

$$\begin{gathered} a^2-b^2+a-b=(a^2-b^2)+(a-b) [Grouping the terms] \\ =(a+b)(a-b)+(a-b) \\ =(a-b)(a+b+1) [Taking out the common factor (a-b\left)\right] \end{gathered}$$

Question 40:

Factorize each of the following expression:
16a⁴ − b⁴

Answer 40:

$$\begin{gathered} 16a^4-b^4 \\ =(4a^2{)}^2-(b^2{)}^2 \\ =(4a^2+b^2)(4a^2-b^2) \\ =(4a^2+b^2)\left[\right(2a{)}^2-b^2] \\ =(4a^2+b^2\left)\right(2a+b\left)\right(2a-b) \end{gathered}$$

Question 41:

Factorize each of the following expression:
a⁴ − 16(b − c)⁴

Answer 41:

$$\begin{gathered} a^4-16(b-c{)}^4 \\ =(a^2{)}^2-\left[4\right(b-c{)}^2{]}^2 \\ =[a^2+4(b-c{)}^2\left]\right[a^2-4(b-c{)}^2] \\ =[a^2+4(b-c{)}^2\left]\right\{a^2-\left[2\right(b-c\left){]}^2\right\} \\ =[a^2+4(b-c{)}^2\left]\right[a+2(b-c)\left]\right[a-2(b-c)] \\ =[a^2+4(b-c{)}^2](a+2b-2c)(a-2b+2c) \end{gathered}$$

Question 42:

Factorize each of the following expression:
2a⁵ − 32a

Answer 42:

$$\begin{gathered} 2a^5-32a \\ =2a(a^4-16) \\ =2a\left[\right(a^2{)}^2-4^2] \\ =2a(a^2+4\left)\right(a^2-4) \\ =2a(a^2+4\left)\right(a^2-2^2) \\ =2a(a^2+4\left)\right(a+2\left)\right(a-2) \\ =2a(a-2\left)\right(a+2\left)\right(a^2+4) \end{gathered}$$

Question 43:

Factorize each of the following expression:
a⁴b⁴ − 81c⁴

Answer 43:

$$\begin{gathered} a^4{b}^4-81c^4 \\ =(a^2{b}^2{)}^2-(9c^2{)}^2 \\ =(a^2{b}^2+9c^2)(a^2{b}^2-9c^2) \\ =(a^2{b}^2+9c^2)\left[\right(ab{)}^2-\left(3c{)}^2\right] \\ =(a^2{b}^2+9c^2)(ab+3c)(ab-3c) \end{gathered}$$

Question 44:

Factorize each of the following expression:
xy⁹ − yx⁹

Answer 44:

$$\begin{gathered} x{y}^9-y{x}^9 \\ =xy(y^8-x^8) \\ =xy\left[\right(y^4{)}^2-\left(x^4{)}^2\right] \\ =xy(y^4+x^4)(y^4-x^4) \\ =xy(y^4+x^4)\left[\right(y^2{)}^2-\left(x^2{)}^2\right] \\ =xy(y^4+x^4)(y^2+x^2)(y^2-x^2) \\ =xy(y^4+x^4)(y^2+x^2)(y+x)(y-x) \end{gathered}$$

Question 45:

Factorize each of the following expression:
x³ − x

Answer 45:

$$\begin{gathered} x^3-x \\ =x(x^2-1) \\ =x(x-1)(x+1) \end{gathered}$$

Question 46:

Factorize each of the following expression:
18a²x² − 32

Answer 46:

$$\begin{gathered} 18a^2{x}^2-32 \\ =2(9a^2{x}^2-16) \\ =2\left[\right(3ax{)}^2-4^2] \\ =2(3ax-4\left)\right(3ax+4) \end{gathered}$$