Question 12 Marks
Solve:
$ 27 x^2-12 y^2 $
Answer$ 27 x^2-12 y^2 $
$ =3\left(9 x^2-4 y^2\right) $
$ =3\left[(3 x)^2-(2 y)^2\right] $
$ =3(3 x-2 y)(3 x+2 y) $
View full question & answer→Question 22 Marks
Factorize of the following algebraic expression:
$-4(x-2 y)^2+8(x-2 y)$
Answer$-4(x-2 y)^2+8(x-2 y)$
$= [-4(x - 2y) + 8] (x - 2y)$ [taking $(x 2)$ as the common factor]
$= 4[-(x - 2y) + 2] (x - 2y) [$taking $4$ as the common factor of $[-4(x - 2y) + 8]]$
$= 4(2y - x + 2)(x -2y)$
View full question & answer→Question 32 Marks
Factories:
$x^2- 4x - 21$
AnswerTo factories $x^2- 4x - 21$, we will find two number p and q such that $p + q = -4$ and $pq = -21$
Now,
$3 + (-7) = -4$ And $3 x (-7) = -21$
Splittiong the middle term 14a in the given quadratic as $-7x + 3x,$ we get:
$x^2- 4x - 21 = x^2- 7x + 3x - 21$
$= (x2 - 7x) + (3x - 21)$
$= x(x - 7) + 3(x - 7)$
$= (x - 7)(x + 3)$
View full question & answer→Question 42 Marks
Factorize of the following expressions:$ 4(x y-1)^2-9(x-1)^2 $
Answer$ 4(x y-1)^2-9(x-1)^2 $
$ =[2(x y+1)]^2-[3(x-1)]^2 $
$ =[2(x y+1)-3(x-1)][2(x y+1)+3(x-1)] $
$ =(2 x y+2-3 x+3)(2 x y+2+3 x-3) $
$ =(2 x y-3 x+5)(2 x y+3 x-1) $
View full question & answer→Question 52 Marks
Solve: $36a^2+ 36a + 9$
Answer$36a^2+ 36a + 9$
$= 9(4a^2+ 4a + 1) = 9{(2a)^2+ 2 × 2a × 1 + 1^2}$
$= 9(2a + 1)^2$
$= 9(2a + 1)(2a + 1)$
View full question & answer→Question 62 Marks
Factorize of the following expressions:$16(2x - 1)^2- 25y^2$
Answer$ 16(2 x-1)^2-25 y^2 $
$ =[4(2 x-1)]^2-(5 y)^2 $
$ =[4(2 x-1)-5 y][4(2 x-1)+5 y] $
$ =(8 x-4-5 y)(8 x-4+5 y) $
$ =(8 x-5 y-4)(8 x+5 y-4) $
View full question & answer→Question 72 Marks
Solve:
$ 3 a^5-48 a^3 $
Answer$ 3 a^5-48 a^3 $
$ =3 a^3\left(a^2-16\right) $
$ =3 a^3\left(a^2-4\right) $
$ =3 a^3(a-4)(a+4) $
View full question & answer→Question 82 Marks
Solve:
$ 64-(a+1)^2 $
Answer$ 64-(a+1)^2 $
$ =(8)^2-(a+1)^2 $
$ =[8-(a+1)][8+(a+1)] $
$ =(8-a-1)(8+a+1) $
$ =(7-a)(9+a) $
View full question & answer→Question 92 Marks
Solve:
$ x^5-16 x^3 $
Answer$ x^5-16 x^3 $
$ =x^3\left(x^2-16\right) $
$ =x^3\left(x^2-4^2\right) $
$ =x^3(x-4)(x+4) $
View full question & answer→Question 102 Marks
Factorize of the following algebraic expression:
$6(a+2 b)-4(a+2 b)^2$
Answer$6(a+2 b)-4(a+2 b)^2$
$= [6 - 4(a + 2b) [$taking $(a + 2b$ as the common factor$]$
$= 2[3 - 2(a + 2b)] [$taking $2$ as the common factor of $[6 - 4(a + 2b)]]$
$= 2(3 - 2a - 4b)(a + 2b)$
View full question & answer→Question 112 Marks
Solve:
$36 L^2-(m+n)^2$
Answer$36 L^2-(m+n)^2$
$ =(6 L)^2-(m+n)^2 $
$ =[6 L-(m+n)][6 L+(m+n)] $
$ =(6 L-m-n)(6 L+m+n)$
View full question & answer→Question 122 Marks
Factorize:
$2x^3y^2- 4x^2y^3+ 8xy^4$
AnswerThe greatest common factor of the terms
$2 x^3 y^2,-4 x^2 y^3$ and $8 x y^4$ of the expression $2 x^3 y^2-4 x^2 y^3+8 x y^4 y^{64}$ is $2 x y^2$
Now,
$2 x^3 y^2=2 x y^2 \cdot x^2-4 x^2 y^3=2 x y^2 \cdot-2 x y 8 x y^4=2 x y^2 \cdot 4 y^2$
Hence, the expression $2 x^3 y^2-4 x^2 y^3+8 x y^4$ can be factorised as $2 x y^2\left(x^2-2 x y+4 y^2\right)$
View full question & answer→Question 132 Marks
Factorize of the following expressions:$x - y - x^2+ y^2$
Answer$x - y - x^2+ y^2$
$= (x - y) + (y^2- x^2)$
$= (x - y) + (y +x)(y - x)$
$= (x - Y) - (y + x)(x - y) [$since, $(y - x) = -(x - y)]$
$= (x - y)[1 - (y + x)$
$= (x - y)(1 - x - y)$
View full question & answer→Question 142 Marks
Factorize of the following algebraic expression: $4(x + y)(3a - b) + 6(x + y)(2b - 3a)$
Answer$4(x + y)(3a - b) + 6(x + y)(2b - 3a)$
$= 2(x + y) [2(3a - b) + 3(2b - b)] [$taking $(2(x + y))$ as the common factor$] $
$= 2(x + y)(6a - 2b + 6b - 9a) = 2(x - y)(4b - 3a)$
View full question & answer→Question 152 Marks
Find the greatest common factor of the polynomial: $x^3, -yx^2$
AnswerThe common literal appearing in the two monomials is $x.$
The smallest power of $x$ in both the monomials is $2. $
Hence, the greatest common factor is $x^2$.
View full question & answer→Question 162 Marks
Solve:
$(x+2 y)^2-4(a x-y)^2$
Answer$(x+2 y)^2-4(a x-y)^2$
$= [(x + 2y) - 2(2x - y)][(x + 2y) + 2(2x - y)]$
$= (x + 2y - 4x + 2y)(x + 2y + 4x - 2y)$
$= 5x(4y - 3x)$
View full question & answer→Question 172 Marks
Factorize of the following expressions:
$a b-b y-a y+y^2$
Answer$a b-b y-a y+y^2$
$= (ab - ay) + (y^2- by)$
$= a(b - y) + y(y - b) [$since, $(y - b) = -(b - y)]$
$= a(b - y) - y(b - y) [$taking $(b - y)$ as the common factor$]$
$= (a -y)(b - y)$
View full question & answer→Question 182 Marks
Factorize of the following expressions: $x^2- 11xy - x + 11y$
Answer$x^2-11 x y-x+11 y=\left(x^2-x\right)+(11 y-11 x y)$
$= x(x - 1) + 11y(1 - x)$
$= x(x - 1) - 11y(x - 1) [$since,$ (1 - x) = -(x - 1)$
$= (x - 11y)(x - 1) [$taking out the common factor$]$
View full question & answer→Question 192 Marks
Factorize of the following expressions: $16(a - b)^3- 24 (a - b)^2$
Answer $16(a-b)^3-24(a-b)^2$
$=8(a-b)^2[2(a-b)-3]\left[\right.$ taking $8(a-b)^2$ as the common factor$]$
$=8(a-b)^2(2 a-2 b-3)$
View full question & answer→Question 202 Marks
Solve:
$ x^4-625 $
Answer$ x^4-625 $
$ =\left(x^2\right)^2-25^2 $
$ =\left(x^2+25\right)\left(x^2-25\right) $
$ =\left(x^2+25\right)\left(x^2-5^2\right) $
$ =\left(x^2+25\right)(x+5)(x-5) $
View full question & answer→Question 212 Marks
Solve:
$ (x+2)^2-6(x+2)+9 $
Answer$ (x+2)^2-6(x+2)+9 $
$ =(x+2)^2-2 \cdot(x+2) \cdot 3+3^3 $
$ =[(x+2)-3]^2 $
$ =(x+2-3)^2 $
$ =(x-1)^2 $
$ =(x-1)(x-1) $
View full question & answer→Question 222 Marks
Factorize of the following expressions: $qr - pr + qs -ps$
Answer$qr - pr + qs -ps $
$= (qr - pr) + (qs - ps) $
$= r(q - p) + s(q - p)$
$= (r + s)(q - p) [$taking $(q - p)$ as the common factor$]$
View full question & answer→Question 232 Marks
Solve:
$(x+y a)^2-(a-b)^2$
Answer$(x+y a)^2-(a-b)^2$
$= [(x + y) - (a - b)][(x + y) + (a - b)]$
$= (x + y - a + b)(x + y + a - b)$
View full question & answer→Question 242 Marks
Factorize of the following algebraic expressions: $5(x - 2y) + 3(x - 2y)$
Answer$5(x - 2y) + 3(x - 2y)$
$= [(x - 2y) + 3] (x - 2y) [$taking $(x - 2y)$ as the commom factor$]$
$= (5x - 10y + 3)(x - 2y)$
View full question & answer→Question 252 Marks
Solve:
$ a^2-b^2+2 b c-c^2 $
Answer$ a^2-b^2+2 b c-c^2 $
$ =a^2-\left(b^2-2 b c+c^2\right) $
$ =a^2-\left(b^2-2 \times b \times c+c^2\right) $
$ =a^2-(b-c)^2 $
$ =[a-(b-c)][a+(b-c)] $
$ =(a-b+c)(a+b-c) $
View full question & answer→Question 262 Marks
Solve:
$ 16 a^2-b^4 $
Answer$ 16 a^2-b^4 $
$ =\left(4 a^2\right)^2-\left(b^2\right)^2 $
$ =\left(4 a^2+b^2\right)\left(4 a^2-b^2\right) $
$ =\left(4 a^2+b^2\right)\left[(2 a)^2-b^2\right] $
$ =\left(4 a^2+b^2\right)(2 a+b)(2 a-b) $
View full question & answer→Question 272 Marks
Solve:
$ x^2-y^2+6 y-9 $
Answer$ x^2-y^2+6 y-9 $
$ =x^2-\left(y^2+6 y-9\right) $
$ =x^2-\left(y^2-2\times y \times 3+3^2\right) $
$ =x^2-(y-3)^2 $
$ =[x-(y-3)][x+(y-3)] $
$ =(x-y+3)(x+y-3) $
View full question & answer→Question 282 Marks
Solve:
$ 144 a^2-169 b^2 $
Answer$ 144 a^2-169 b^2 $
$ =(12 a)^2-\left(13 b^2\right) $
$=(12 a-13 b)(12 a+13 b) $
View full question & answer→Question 292 Marks
Solve:
$ (2 a-b)^2-16 c^2 $
Answer$ (2 a-b)^2-16 c^2 $
$ =(2 a-b)^2-(4 c)^2 $
$ =[(2 a-b)-4 c][(2 a-b)+4 c] $
$ =(2 a-b-4 c)(2 a-b+4 c) $
View full question & answer→Question 302 Marks
Solve:
$(x-4 y)^2-625$
Answer$(x-4 y)^2-625$
$=(x-4 y)^2-25^2$
$= [(x - 4y) - 25][(x - 4y) + 25]$
$= (x - 4y - 25)(x - 4y + 25)$
View full question & answer→Question 312 Marks
Solve:
$ p^2 q^2-p^4 q^4 $
Answer$ p^2 q^2-p^4 q^4 $
$ =p^2 q^2\left(1-p^2 q^2\right) $
$ =p^2 q^2\left[1-(p q)^2\right] $
$ =p^2 q^2(1-p q)(1+p q) $
View full question & answer→Question 322 Marks
Solve:
$ 4 x^4+y^4 $
Answer$ 4 x^4+y^4 $
$ =4 x^4+4 x^2+y^4-4 x^2 y^2 $
$ =\left[\left(2 x^2\right)^2+2 x 2 x^2 x y+\left(y^2\right)^2\right]-(2 x y)^2 $
$ =\left(2 x^2+y^2\right)^2-(2 x y)^2 $
$ =\left[\left(2 x^2+y\right)-2 x y\right]\left[2\left(2 x+y^2\right)+2 x y\right] $
$ =\left(2 x^2-2 x y+y 2\right)\left(2 x^2+2 x y+y^2\right) $
View full question & answer→Question 332 Marks
Solve:
$ 4 x^2+12 x y+9 y^2 $
Answer$ 4 x^2+12 x y+9 y^2 $
$ =(2 x)^2+2 \times 2 x \times 3 y+(3 y)^2 $
$ =(2 x+3 y)^2 $
$ =(2 x+3 y)(2 x+3 y) $
View full question & answer→Question 342 Marks
Factorize:
$9x^2y + 3axy$
AnswerThe greatest common factor of the term $9x^2y$ and 3axy of the expression
Also, we can write $9x^2y = 3axy.3x$ and $3axy = 3xy.a$
Therefore, $9x^2y + 3axy = (3xy.3x) + (3xy.a)$
$= 3xy(3x + a)$
View full question & answer→Question 352 Marks
Solve:
$xy^9- yx^9$
Answer$xy^9- yx^9$
$ =x y\left(y^8-x^8\right) $
$ =x y\left[\left(y^4\right)^2-\left(x^4\right)^2\right] $
$ =x y\left(y^4+x^4\right)\left[\left(y^2\right)^2-\left(x^2\right)^2\right] $
$ =x y\left(y^4+x^4\right)\left(y^2+x^2\right)\left(y^2-x^2\right) $
$ =x y\left(y^4+x^4\right)\left(y^2+x^2\right)(y+x)(y-x) $
View full question & answer→Question 362 Marks
Solve:
$ 16 x^2-25 y^2$
Answer$ 16 x^2-25 y^2$
$ =(4 x)^2-(5 y)^2 $
$ =(4 x-5 y)(4 x+5 y) $
View full question & answer→Question 372 Marks
Solve:
$ a^4+3 a^2+4 $
Answer$ a^4+3 a^2+4 $
$ =a^4+4 a^2-a^2+4 $
$ =\left(a^4+4 a^2+4\right)-a^2 $
$ =\left[\left(a^2\right)^2+2 \times a^2 \times 2+2^2\right]-a^2 $
$ =\left(a^2+2\right)^2-a^2 $
$ =\left[\left(a^2+2\right)-a\right]\left[\left(a^2+2\right)+a\right] $
$ =\left(a^2-a+2\right)\left(a^2+a+2\right) $
View full question & answer→Question 382 Marks
Factorize of the following expressions: $ax + ay - bx - by$
Answer$ax + ay - bx - by$
$= (ax + ay) - (bx + by)$
$= a(x + y) - b(x + y)$
$= (a - b)(x + y) [$taking $(x + y)$ as the common factor$]$
View full question & answer→Question 392 Marks
Solve:
$ a^4-(2 b+c)^4 $
Answer$ a^4-(2 b+c)^4 $
$ =\left(a^2\right)^2-\left[(2 b+c)^2\right]^2 $
$ =\left[a^2+(2 b+c)^2\right]\left[a^2-(2 b+c)^2\right] $
$ =\left[a^2+(2 b+c)^2\right]\{[a+(2 b+c)][a-(2 b+c)]\} $
$ =\left[a^2+(2 b+c)^2\right](a+2 b+c)(a-2 b-c) $
View full question & answer→Question 402 Marks
Factories:
$a^2+ 14a + 48$
AnswerTo factories $a^2+ 14a + 48$, we will find two number $p$ and $q$ such that $p + q = 14$ and $pq = 48$
$8 + 6 = 14$ And $8 x 6 = 48$
Splittiong the middle term $14a$ in the given quadratic as $8a + 6a,$ we get:
$a^2+ 14a + 48 =a^2+ 8a + 6a + 48$
$= (a2 + 8a) + (6a + 48)$
$= a(a + 8) + 6(a + 8)$
$= (a + 6)(a + 8)$
View full question & answer→Question 412 Marks
Factorize: $3x - 9$
AnswerThe greatest common factor of the terms $3x$ and $-9$ of the expression $3x - 9$ is $3$ Now, $3x = 3x$ and $-9 = 3(-3)$ Hence, the expression $3x - 9$ can be factorised as $3(x - 3)$
View full question & answer→Question 422 Marks
Solve:
$ 144 a^2-289 b^2$
Answer$ 144 a^2-289 b^2$
$ =(12 a)^2-(17 b)^2 $
$ =(12 a-17 b)(12 a+17 b) $
View full question & answer→Question 432 Marks
Solve:
$ a^4 b^4-16 c^4 $
Answer$ a^4 b^4-16 c^4 $
$ =\left[\left(a^2 b^2\right)^2-\left(4 c^2\right)^2\right] $
$ =\left(a^2 b^2+4 c^2\right)\left(a^2 b^2-4 c^2\right) $
$ =\left(a^2 b^2+4 c^2\right)\left[(a b)^2-(2 c)^2\right]$
$ =\left(a^2 b^2+4 c^2\right)(a b+2 c)(a b-2 c) $
View full question & answer→Question 442 Marks
Solve:
$125 x^2-45 y^2 $
Answer$125 x^2-45 y^2 $
$ =5\left(25 x^2-9 y^2\right) $
$ =5\left[(5 x)^2-(3 y)^2\right] $
$ =5(5 x-3 y)(5 x+3 y) $
View full question & answer→Question 452 Marks
Solve:
$ x^4-144 x $
Answer$ x^4-144 x $
$ =x\left(x^2-144\right) $
$ =x\left(x^2-12^2\right) $
$ =x(x-12)(x+12) $
View full question & answer→Question 462 Marks
Factorize: $x^4y^2- x^2y^4- x^4y^4$ is $x^2y^2$
AnswerThe greatest common factor of the term
$x^4 y^2-x^2 y^4$ and $x^4 y^4$ of the expression
$x^4 y^2-x^2 y^4-x^4 y^4$ is $x^2 y^2$
Also, we can write $x^4 y^2=x^2 y^4\left(x^2 y^2 \cdot x^2\right), x^2 y^4=\left(x^2 y^2 \cdot y^2\right)$ and $x^4 y^4=\left(x^2 y^2 \cdot x^2 y^2\right)$
Therefore, $x^4 y^2-x^2 y^4-x^4 y^4=\left(x^2 y^2 \cdot x^2\right)-\left(x^2 y^2 \cdot y^2\right)-\left(x^2 y^2 \cdot x^2 y^2\right)$
$=x^2 y^2\left(x^2-y^2-x^2 y^2\right)$
View full question & answer→Question 472 Marks
Solve:
$a^2-b^2+a-b$
Answer$a^2-b^2+a-b$
$= (a + b)(a -b) + (a - b)$
$= (a - b)(a + b + 1)$
View full question & answer→Question 482 Marks
Factorize of the following expressions:
$abx^2+ (ay - b) x - y$
Answer$abx^2+ (ay - b) x - y$
$= (abx^2- bx) + (axy - y)$
$= bx(ax -1) + y(ax - 1)$
$= (bx + y)(ax -1)[$taking $(ax - 1)$ as the common factor$]$
View full question & answer→Question 492 Marks
Factorize of the following expressions: $x^3-2 x^2 y+3 x y^2-6 y^3 $
Answer$x^3-2 x^2 y+3 x y^2-6 y^3 $
$ =\left(x^3-2 x^2 y\right)+\left(3 x y^2-6 y^3\right) $
$ =x^2(x-2 y)+3 y^2(x-2 y) $
$ =\left(x^2+3 y^2\right)(x-2 y) [$taking $(x - 2y)$ as the common factor$]$
View full question & answer→Question 502 Marks
Solve:
$ 9 a^2-24 a b+16 b^2 $
Answer$ 9 a^2-24 a b+16 b^2 $
$ =(3 a)^2-2 \times 3 a \times 4 b+(4 b)^2 $
$ =(3 a-4 b)^2 $
$ =(3 a-4 b)(3 a-4 b) $
View full question & answer→