Question 512 Marks
Solve:
$ 3 x^3 y-243 x y^3 $
Answer$ 3 x^3 y-243 x y^3 $
$ =3 x y\left(x^2-81 y^2\right) $
$ =3 x y\left[x^2-(9 y)^2\right] $
$ =3 x y(x-9)(x+9 y) $
View full question & answer→Question 522 Marks
Solve:
$ x^4-(2 y-3 z)^2 $
Answer$ x^4-(2 y-3 z)^2 $
$ =(x 2)^2-(2 y-3 z)^2 $
$ =[x 2-(2 y-3 z)][x 2+(2 y-3 z)] $
$ =(x 2-2 y+3 z)(x 2+2 y-3 z) $
View full question & answer→Question 532 Marks
Solve:
$ a^4-1 / b^4 $
Answer$ a^4-1 / b^4 $
$ =\left(a^2\right)^2-1 /\left(b^2\right)^2 $
$ =a^2-1 / b^2 a^2+1 / b^2 $
$ =a-1 / b a+1 / b a^2+1 / b^2 $
View full question & answer→Question 542 Marks
Solve:
$ x^4-1 $
Answer$ x^4-1 $
$ =\left(x^2\right)^2-1 $
$ =\left(x^2+1\right)\left(x^2-1\right) $
$ =\left(x^2+1\right)\left(x^2-1\right) $
$ =\left(x^2+1\right)(x+1)(x-1) $
View full question & answer→Question 552 Marks
Factorize of the following expressions: $6xy + 6 - 9y - 4x$
Answer$6xy + 6 - 9y - 4x$
$= 2x(3y - 2) + 3(2 - 3y)$
$= 2x(3y - 2) - 3(3y - 2) [$since, $(2 - 3y) = -(3y - 2)]$
$= 2x(3y - 3)(3y - 2)$
$[$taking $(3y - 2)$ as the common factor$]$
View full question & answer→Question 562 Marks
Factorize:
$72 x^6 y^7-96 x^7 y^6$
AnswerThe greatest common factor of the terms
$72 x^6 y^7$ and $-96 x^7 y^6$ of the expression $72 x^6 y^7-96 x^7 y^{64}$ is $24 x^6 y^6$
Now,
$72 x^6 y^7=24 x^6 y^6 \cdot 3 y \text { and }-96 x^7 y^6=24 x^6 y^6 \cdot-4 x$
Hence, the expression $72 x^6 y^7-96 x^7 y^6$ can be factorised as $24 x^6 y^6 \cdot(3 y-4 x)$
View full question & answer→Question 572 Marks
Factorize of the following algebraic expression:
$x^3(a-2 b)+x^2(a-2 b)$
Answer$x^3(a-2 b)+x^2(a-2 b)$
$= (x^3+ x^2)(a - b)$ [taking $(a - 2b)$ as the common factor]
$= x^2(x + 1)(a - 2b) [$taking $x2$ as the common factor of $(x^3+ x^2)]$
View full question & answer→Question 582 Marks
Factorize of the following expressions:
$x a^2+x b^2-y a^2-y b^2$
Answer$ x a^2+x b^2-y a^2-y b^2 $
$ =\left(x a^2+x b^2\right)-\left(y a^2-y b^2\right) $
$ =x\left(a^2+b^2\right)-y\left(a^2-b^2\right) $
$= (x - y)(a^2- b^2)$ [taking $(a^2- b^2)$ as the common factor]
View full question & answer→Question 592 Marks
Solve:
$4x^4+ 1$
Answer$4x^4+ 1$
$= 4x^4+ 4x^2+ 1 - 4x^2$
$= [(2x)^2+ 2 x 2x^2 x 1 + 1] - 4x^2$
$= (2x^2+ 1) - (2x)^2$
$= [(2x^2+ 1) - 2x][(2x^2+ 1) + 2x]$
$= (2x^2- 2x + 1)(2x^2+ 2x + 1)$
View full question & answer→Question 602 Marks
Factorize:
$20 a^{12} b^2-15 a^8 b^4$
AnswerThe greatest common factor of the terms
$20 a^{12} b^2$ and $-15 a^8 b^4$ of the expression $20 a^{12} b^2-15 a^8 b^4$ is $5 a^8 b^2$.
$20 a^{12} b^2=5 \times 4 \times a^8 \times a^4 \times b^2=5 a^8 \times b^2 \times 4 a^4$ and $-15 a^8 b^4=5 \times-3 \times a^8 \times b^2 \times b^2=5 a^8 b^2 \times(-3) b^2$
Hence, the expression $20 a^{12} b^2-15 a^8 b^4$ can be factorised as $5 a^8 b^2\left(4 a^4-3 b^2\right)$
View full question & answer→Question 612 Marks
Factorize:
$16m - 4m^2$
AnswerThe greatest common factor of the term
$16m$ and $4m^2$ of the expression
Also, we can write $16m - 14m.4$ and $4m^2= 4m.m$
Therefore, $16m - 4m^2= (4m.4) - (4m.m)$
$= 4m(4 - m)$
View full question & answer→Question 622 Marks
Factorize of the following expressions: $a(a + b - c) - bc$
Answer$a(a + b - c) - bc= a^2+ ab - ac - bc$
$= (a^2- ac) + (ab - bc)$
$= a(a - c) + b(a - c)$
$= (a + b)(a - c) [$taking $(a - c)$ as the common factor$]$
View full question & answer→Question 632 Marks
Factorize of the following algebraic expressions:
$9a(6a - 5b) - 12a^2(6a - 5b)$
Answer$9a(6a - 5b) - 12a^2(6a - 5b)$
$= (9a -12qa^2)(6a - 5a)$ [taking $(6a - 5b)$ as the common factor]
$= 3a(3 - 4)(6a - 5b) [$taking 3a as the common factor of the quadratic eqn. $(9a - 12a^2)]$
View full question & answer→Question 642 Marks
Factorize of the following expressions:
$a^2 x^2+\left(a x^2+1\right) x+a$
Answer$a^2 x^2+\left(a x^2+1\right) x+a$
$=\left(a x^3+a^2 x^2\right)+(x+a)$
$=a x^2(x+a)+(x+a)$
$=\left(a x^2+a\right)(x+a)[$taking $(x +a )$ as the common factor$]$
View full question & answer→Question 652 Marks
Solve:
$ 12 m^2-27 $
Answer$ 12 m^2-27 $
$ =3\left(4 m^2-9\right) $
$ =3\left[(2 m)^2-3^2\right] $
$ =3(2 m-3)(2 m+3) $
View full question & answer→Question 662 Marks
Factorize of the following expressions: $a(a - 2b - c) + 2bc$
Answer$a(a - 2b - c) + 2bc= a^2- 2ab - ac + 2bc$
$= (a^2- ac) + (2bc - 2ab)$
$= a(a - c) + 2b(c - a) $[since, $(9c - a)= -(a - c)]$
$= a(a - c) - 2b(a - c)$
$= (a - 2b)(a - c)$ [taking $(a - c)$ as the common factor]
View full question & answer→Question 672 Marks
Solve:
$(3 - 2a)^2- 25a^2$
Answer$(3 - 2a)^2- 25a^2$
$= (3 + 2a)^2- 25a^2$
$= [(3a + 2a) - 5a][(3 + 2a) + 5a]$
$= (3 + 2a - 5a)(3 + 2a + 5a)$
$= (3 - 3a)(3 + 7a)$
$= 3(1 - a)(3 + 7a)$
View full question & answer→Question 682 Marks
Factorize of the following expressions:
$ab - a - b + 1$
Answer$ab - a - b + 1$
$= (ab - b) + (1 - a)$
$= b(a - 1) + (1 - a)$
$= b(a - 1) - (a - 1) [$since, $(1 - a) = -(a -1)]$
$= (a - 1)(b - 1) [$taking out the common factor $(a - 1)]$
View full question & answer→Question 692 Marks
Factorize of the following expressions:
$x^2+ y - xy - x$
Answer$x^2+y-x y-x=\left(x^2-x y\right)+(y-x)$
$= x(x - y) + (y - x)$
$= x(x - y) - (x - y) [(y - x)$
$= -(x - y)] = (x - 1)(x - y)$ [taking $(x - y)$ as the common factor]
View full question & answer→Question 702 Marks
Solve:
$ a^2+2 a b+b^2-16 $
Answer$ a^2+2 a b+b^2-16 $
$ =a^2+2 \times a \times b+b^2-16 $
$ =(a+b)^2-4^2 $
$ =(a+b-4)(a+b+4) $
View full question & answer→Question 712 Marks
Factorize of the following expressions:
$ x^3-y^2+x-x^2 y^2 $
Answer$ x^3-y^2+x-x^2 y^2 $
$ =\left(x^3+x\right)-\left(x^2 y^2+y^2\right) $
$ =x\left(x^2+1\right)-y^2\left(x^2+1\right) $
$ =\left(x-y^2\right)\left(x^2+1\right)\left[\text { taking }\left(x^2+1\right) \text { as the common factor] }\right. $
View full question & answer→Question 722 Marks
Solve:
$p^2 q^2-6 q r+9 r^2=(p q)^2-2 \times p q \times 3 r+(3 r)^2 $
Answer$p^2 q^2-6 q r+9 r^2=(p q)^2-2 \times p q \times 3 r+(3 r)^2 $
$ =(p q-3 r)^2 $
$ =(p q-3 r)(p q-3 r) $
View full question & answer→Question 732 Marks
Solve:
$ 25 x^4 y^4-1 $
Answer$ 25 x^4 y^4-1 $
$ =\left(5 x^2 y^2\right)^2-1$
$ =\left(5 x^2 y^2-1\right)\left(5 x^2 y^2+1\right) $
View full question & answer→Question 742 Marks
Factorize of the following expressions: $axy + bcxy - az - bcz$
Answer$axy + bcxy - az - bcz$
$= (axy + bcxy) - (az - bcz)= xy(a + bc) - z(a + bc)$
$= (xy - z)(a + bc) [$taking $(a + bc)$ as the common factor$]$
View full question & answer→Question 752 Marks
Factorize:
$-4a^2+ 4ab - 4ca$
AnswerThe greatest common factor of the term
$-4a^2+ 4ab$ and 4ca of the expression
$-4a^2+ 4ab - 4ca$
Also, we can write $-4a^2= (-4a.a), 4ab = -4a.(-b) - 4ca$ and $4ca = (-4a.c)$
Therefore, $-4a^2+ 4ab - 4ca = (-4a.a) + (-4a.(-b)) - (4a.c)$
$-4a(a - b)$
View full question & answer→Question 762 Marks
Factories:
$y^2+ 5y - 36$
AnswerTo factories $y^2+ 5y - 36$, we will find two number p and q such that $p + q = 5$ and $pq = -36$
Now,
$9 + (-4) = 5$
And
$9 x (-4) = -36$
Splittiong the middle term 5y in the given quadratic as $-4y + 9y,$ we get:
$y^2+ 5y - 36 = y^2-4y +9y - 36$
$= (y2 - 4y) + (9y - 36)$
$= y(y - 4) + 9(y - 4)$
$= (y + 9)(y - 4)$
View full question & answer→Question 772 Marks
Factorize:
$x^2yz + xy^2z + xyz^2$
AnswerThe greatest common factor of the term
$x^2yz, xy^2z$ and $xyz^2$ of the expression
$x^2yz + xy^2z + xyz^2 $ is $ xyz$.
Also, we can write$x^2yz = (xyz.x), (xy^2z = xyz.y), xy^2z = (xyz.z)$
Therefore, $x^2yz + xy^2z + xyz^2= (xyz.x) + (xyz.y) + (xyz.z)$
View full question & answer→Question 782 Marks
Factorize of the following expressions:
$Lm^2- mn^2- Lm + n^2$
Answer$ L m^2-m n^2-L m+n^2=\left(L m^2-L m\right)+\left(n^2-m n^2\right) $
$ ={Lm}(m-1)+n^2(1-m) $
$ ={Lm}(m-1)-n^2(m-1)[\text { since, }(1-m) $
$= -(m - 1)] = (Lm - n^2)(m - 1) [$taking $(m -1)$ as the common factor$]$
View full question & answer→Question 792 Marks
Factorize of the following expressions: $1 + x + xy + x^2y$
Answer$1 + x + xy + x^2y$
$= (1 + x) + (xy + x^2y)$
$= (1 + x) + xy(1 + x)$
$= (1 + xy)(1 + x) [$taking $(1 + x)$ as the common factor$]$
View full question & answer→Question 802 Marks
Solve:
$ 75 a^3 b^2-108 a b^4 $
Answer$ 75 a^3 b^2-108 a b^4 $
$ =3 a b^2\left(25 a^2-36 b^2\right) $
$ =3 a b^2\left[(5 a)^2-(6 B)^2\right] $
$ =3 a b^2(5 a-6 b)(5 a+6 b) $
View full question & answer→Question 812 Marks
Factorize: $a x^2 y+b x y^2+c x y z$
AnswerThe greatest common factor of the term
$a x^2 y+b x y^2$ and $cxyz$ of the expression
$a x^2 y+b x y^2+c x y z \text { is } x y$
Also,we can write $a x^2 y=(x y \cdot a x), b x y^2=(x y . b y), c x y z=(x y . c z)$
Therefore, $a x^2 y+b x y^2+c x y z=(x y \cdot a x)+(x y \cdot b y)+(x y \cdot c z)=x y(a x+b y+c z)$
View full question & answer→Question 822 Marks
Solve:
$ 9(a-b)^2-100(x-y)^2 $
Answer$ 9(a-b)^2-100(x-y)^2 $
$ =[3(a-b)]^2-[10(x-y)]^2 $
$= [3(a - b) - 10(x - y)][3(a - b) + 10(x - y)]$
$= (3a - 3b - 10x + 10y)(3a - 3b + 10x - 10)$
View full question & answer→Question 832 Marks
Factorize: $5x - 15x^2$
AnswerThe greatest common factor of the terms $5x$ and $15x^2$ of the expression $5x - 15x^2$ is $5x$
Now,
$5x = 5x.(-1)$ and $-15x^2= 5x.(-3x)$
Hence, the expression $5x - 15x^2$ can be factorised as $5x(1 - 3x)$
View full question & answer→Question 842 Marks
Solve: $\frac{50}{(\text{x})^2}-\frac{2\text{x}^2}{81}$
Answer$\frac{50}{(\text{x})^2}-\frac{2\text{x}^2}{81}$ $=2\Big(\frac{25}{(\text{x})^2}-\frac{\text{x}^2}{81}\Big)$ $=2\Big\{\frac{25}{(\text{x})^2}-\frac{2\text{x}^2}{81}\Big\}$ $=2\Big(\frac{5}{\text{x}}-\frac{\text{x}}{9}\Big)\Big(\frac{5}{\text{x}}+\frac{\text{x}}{9}\Big)$
View full question & answer→Question 852 Marks
Solve:
$144 a^2-289 b^2 $
Answer$144 a^2-289 b^2 $
$= (12 a)^2-(17 b)^2 $
$= (12a - 17b)(12a + 17b)$
View full question & answer→Question 862 Marks
Solve:
$a^2+2 a b+b^2-c^2$
View full question & answer→Question 872 Marks
Factorize of the following expressions: $2ax + bx + 2ay + by$
Answer$2ax + bx + 2ay + by$
$= (2ax + bx) + (2ay + by)$
$= x(2a + b) + y(2a + b)$
$= (x + y)(2a + b) [$taking $(2a + b)$ as the common factor$]$
View full question & answer→Question 882 Marks
Solve:
$ 18 a^2 x^2-32 $
Answer$ 18 a^2 x^2-32 $
$ =2\left(9 a^2 x^2-16\right) $
$ =2\left[(3 a x)^2-4^2\right] $
$ =2\left[(3 a x)^2-4^2\right] $
$= (3ax - 4)(3ax + 4)$$
View full question & answer→Question 892 Marks
Factorize of the following expressions:
$x^2-2 a x-2 a b+b x$
Answer$x^2-2 a x-2 a b+b x$
$= (x^2- 2ax) + (bx - 2ab)$
$= x(x - 2a) + b(x - 2a)$
$= (x + b)(x - 2a)$ [taking $(x - 2a)$ as the common factor]
$= (x - 2a)(x + b)$
View full question & answer→Question 902 Marks
Factorize of the following algebraic expression:$(2x - 3y)(a + b) + (3x - 2y)(a + b)$
Answer$(2x - 3y)(a + b) + (3x - 2y)(a + b)$
$ = (2x - 3y + 3x -2y)(a + b) [$taking $(a + b)$ as the common factor$]$
$= (5x - 5y)(a + b)$
$= 5(x - y)(a +b) [$taking $5$ as the common factor of $(5x -5y)]$
View full question & answer→Question 912 Marks
Solve:
$ a^4-16 b^4$
Answer$ a^4-16 b^4$
$=a^4-2^4 b^4$
$ =\left(a^2\right)^2-\left(2^2 b^2\right)^2 $
$=\left(a^2-2^2 b^2\right)\left(a^2+2^2 b^2\right)$
$ =\left[a^2-(2 b)^2\right]\left(a^2+4 b^2\right) $
$ (a-2 b)(a+2 b)\left(a^2+4 b^2\right) $
View full question & answer→Question 922 Marks
Solve:
$ x^8-1$
Answer$ x^8-1=\left(x^4\right)^2-1^2 $
$ =\left(x^4-1\right)\left(x^4+1\right) $
$ =\left[\left(x^2\right)^2-1^2\right]\left(x^4+1\right) $
$ =\left(x^2-1\right)\left(x^2+1\right)\left(x^4+1\right) $
$ =\left(x^2-1^2\right)\left(x^2+1\right)\left(x^4+1\right) $
$ =(x-1)(x+1)\left(x^2+1\right)\left(x^4+1\right) $
View full question & answer→Question 932 Marks
Solve:
$ a^4 b^4-84 c^4 $
Answer$ a^4 b^4-84 c^4 $
$ =\left(a^2 b^2\right)^2-\left(9 c^2\right)^2 $
$ =\left(a^2 b^2+9 c^2\right)\left(a^2 b^2-9 c^2\right) $
$ =\left(a^2 b^2+9 c^2\right)\left[(a b)^2-(2 c)^2\right] $
$ =\left(a^2 b^2+9 c^2\right)(a b+3 c)(a b-3 c) $
View full question & answer→Question 942 Marks
Factorize: $20 x^3-40 x^2+80 x$
AnswerThe greatest common factor of the terms
$20x^3, -40x^2$ and 80x of the expression $20x^3- 40x^2+ 80x$ is $20x$
Now,
$20 x^3=20 x \cdot x^2-40 x^2=20 x-2 x$ and $80 x=20 x \cdot 4$
Hence, the expression $20x^3- 40x^2+ 80x$ can be factorised as $20x(x^2- 2x + 4)$
View full question & answer→Question 952 Marks
Factorize of the following expressions:
$ a b\left(x^2+1\right)+x\left(a^2+b^2\right) $
Answer$ a b\left(x^2+1\right)+x\left(a^2+b^2\right)=a b x^2+a b+a^2 x+b^2 x $
$ =\left(a b x^2+a^2 x\right)+\left(b^2 x+a b\right) $
$= ax(bx + a) + b(bx + a) = (ax + b)(bx + a) [$taking $(bx + a)$ as the common facotor$]$
View full question & answer→Question 962 Marks
Factorize of the following algebraic expression:
$a(x - y) + 2b(y - x) + c(x - y)^2$
Answer$ba(x-y)+2 b(y-x)+c(x-y)^2 $
$ =a(x-y)-2 b(x-y)+c(x-y)^2[(y-x)=-(x-y) $
$= [a - 2b + c(x - y)] (x - y)$
$= (a - 2b + cx - cy)(x - y)$
View full question & answer→Question 972 Marks
Factorize of the following expressions:
$x^2+ xy + xz + yz$
Answer$x^2+ xy + xz + yz$
$= (x^2+ xy) + (xz + yz)$
$= x(x + y) + z(x + y)$
$= (x + z)(x + y) [$taking $(x + y)$ as the common factor$]$
$= (x + y)(x + z)$
View full question & answer→