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$-4(x-2 y)^2+8(x-2 y)$
$=[-4(x-2 y)+8](x-2 y)$ [taking $(x 2)$ as the common factor]
$=4[-(x-2 y)+2](x-2 y)[$ taking 4 as the common factor of $[-4(x-2 y)+8]]$
$=4(2 y-x+2)(x-2 y)$
View full question & answer→Question 32 Marks
Factories:
$x^2-4 x-21$
AnswerTo factories $x^2-4 x-21$, we will find two number $p$ and $q$ such that $p+q=-4$ and $p q=-21$
Now,
$3+(-7)=-4 \text { And } 3 \times(-7)=-21$
Splittiong the middle term $14 a$ in the given quadratic as $-7 x+3 x$, we get:
$x^2-4 x-21=x^2-7 x+3 x-21$
$=(x 2-7 x)+(3 x-21)$
$=x(x-7)+3(x-7)$
$=(x-7)(x+3)$
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:
$36 a^2+36 a+9$
Answer$36 a^2+36 a+9$
$=9\left(4 a^2+4 a+1\right)=9\left\{(2 a)^2+2 \times 2 a \times 1+1^2\right\}$
$=9(2 a+1)^2$
$=9(2 a+1)(2 a+1)$
View full question & answer→Question 62 Marks
Factorize of the following expressions:
$16(2 x-1)^2-25 y^2$
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$\quad 6(a+2 b)-4(a+2 b)^2$
$=[6-4(a+2 b)[\text { taking }(a+2 b \text { as the common factor }]$
$=2[3-2(a+2 b)][\text { taking } 2 \text { as the common factor of }[6-4(a+2 b)]]$
$=2(3-2 a-4 b)(a+2 b)$
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:
$2 x^3 y^2-4 x^2 y^3+8 x y^4$
View full question & answer→Question 132 Marks
Factorize of the following expressions
$x-y-x^2+y^2$
Answer$x-y-x^2+y^2$
$=(x-y)+\left(y^2-x^2\right)$
$=(x-y)+(y+x)(y-x)$
$=(x-y)-(y+x)(x-y)[\text { since, }(y-x)=-(x-y)]$
$=(x-y)[1-(y+x)$
$=(x-y)(1-x-y)$
View full question & answer→Question 142 Marks
Factorize of the following algebraic expression:
4(x + y)(3a - b) + 6(x + y)(2b - 3a)
Answer4(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:
$x^3,-y x^2$
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$=[(x+2 y)-2(2 x-y)][(x+2 y)+2(2 x-y)]$
$=(x+2 y-4 x+2 y)(x+2 y+4 x-2 y)$
$=5 x(4 y-3 x)$
View full question & answer→Question 172 Marks
Factorize of the following expressions:
$a b-b y-a y+y^2$
Answer$a b-b y-a y+y^2$
$=(a b-a y)+\left(y^2-b y\right)$
$=a(b-y)+y(y-b)[$ 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:
$x^2-11 x y-x+11 y$
Answer$x^2-11 x y-x+11 y=\left(x^2-x\right)+(11 y-11 x y)$
$=x(x-1)+11 y(1-x)$
$=x(x-1)-11 y(x-1)[\text { since, }(1-x)=-(x-1)$
$=(x-11 y)(x-1)[\text { taking out the common factor] }$
View full question & answer→Question 192 Marks
Factorize of the following expressions:
$16(a-b)^3-24(a-b)^2$
Answer$16(a-b)^3-24(a-b)^2$
$=8(a-b)^2[2(a-b)-3]\left[\text { taking } 8(a-b)^2 \text { as the common factor] }\right.$
$=8(a-b)^2(2 a-2 b-3)$
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
Answerqr - 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)
Answer5(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$x^2-y^2+6 y-9$
$=x^2-\left(y^2+6 y-9\right)$
$=x^2-\left(y^2-2 x y \times 3+3^2\right)$
$=x^2-(y-3)^2$
$=[x-(y-3)][x+(y-3)]$
$=(x-y+3)(x+y-3)$
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$(x-4 y)^2-625$
$=(x-4 y)^2-25^2$
$=[(x-4 y)-25][(x-4 y)+25]$
$=(x-4 y-25)(x-4 y+25)$
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:
$9 x^2 y+3 a x y$
AnswerThe greatest common factor of the term
$9 x^2 y$ and $3 a x y$ of the expression
Also, we can write $9 x^2 y=3 a x y .3 x$ and $3 a x y=3 x y \cdot a$
Therefore, $9 x^2 y+3 a x y=(3 x y \cdot 3 x)+(3 x y \cdot a)$
$=3 x y(3 x+a)$
View full question & answer→Question 352 Marks
Solve:
$x y^9-y x^9$
Answer$x y^9-y x^9$
$=x y\left(y^8-x^8\right)$
$=x y\left[\left(y^4\right)^2-\left(x^4\right)^2\right]$
$=x y\left(y^4+x^4\right)\left[\left(y^2\right)^2-\left(x^2\right)^2\right]$
$=x y\left(y^4+x^4\right)\left(y^2+x^2\right)\left(y^2-x^2\right)$
$=x y\left(y^4+x^4\right)\left(y^2+x^2\right)(y+x)(y-x)$
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
Answerax + 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:
$a^2+14 a+48$
AnswerTo factories $a^2+14 a+48$, we will find two number $p$ and $q$ such that $p+q=14$ and $p q=48$
$8+6=14 \text { And } 8 \times 6=48$
Splittiong the middle term $14 a$ in the given quadratic as $8 a+6 a$, we get:
$a^2+14 a+48=a^2+8 a+6 a+48$
$=(a 2+8 a)+(6 a+48)$
$=a(a+8)+6(a+8)$
$=(a+6)(a+8)$
View full question & answer→Question 412 Marks
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:
$x^4 y^2-x^2 y^4-x^4 y^4 \text { is } x^2 y^2$
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:
$a b x^2+(a y-b) x-y$
Answer$a b x^2+(a y-b) x-y$
$=\left(a b x^2-b x\right)+(a x y-y)$
$=b x(a x-1)+y(a x-1)$
$=(b x+y)(a x-1)[$ taking $(a x-1)$ 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$x^3-2 x^2 y+3 x y^2-6 y^3$
$=\left(x^3-2 x^2 y\right)+\left(3 x y^2-6 y^3\right)$
$=x^2(x-2 y)+3 y^2(x-2 y)$
$=\left(x^2+3 y^2\right)(x-2 y)[\text { taking }(x-2 y) \text { as the common factor] }$
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→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
Answer6xy + 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,
$72 x^6 y^7=24 x^6 y^6 .3 y \text { and }-96 x^7 y^6=24 x^6 y^6 .-4 x$
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$x^3(a-2 b)+x^2(a-2 b)$
$=\left(x^3+x^2\right)(a-b)[$ taking $(a-2 b)$ as the common factor]
$=x^2(x+1)(a-2 b)\left[\right.$ taking $x 2$ as the common factor of $\left.\left(x^3+x^2\right)\right]$
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$x a^2+x b^2-y a^2-y b^2$
$=\left(x a^2+x b^2\right)-\left(y a^2-y b^2\right)$
$=x\left(a^2+b^2\right)-y\left(a^2-b^2\right)$
$=(x-y)\left(a^2-b^2\right)\left[\right.$ taking $\left(a^2-b^2\right)$ as the common factor]
View full question & answer→Question 592 Marks
Solve:
$4 x^4+1$
Answer$4 x^4+1$
$=4 x^4+4 x^2+1-4 x^2$
$=\left[(2 x)^2+2 \times 2 x^2 \times 1+1\right]-4 x^2$
$=\left(2 x^2+1\right)-(2 x)^2$
$=\left[\left(2 x^2+1\right)-2 x\right]\left[\left(2 x^2+1\right)+2 x\right]$
$=\left(2 x^2-2 x+1\right)\left(2 x^2+2 x+1\right)$
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 $5 a^8 b^2$.
$20 a^{12} b^2=5 \times 4 \times a^8 \times a^4 \times b^2=5 a^8 \times b^2 \times 4 a^4 \text { and }-15 a^8 b^4=5 \times-3 \times a^8 \times b^2 \times b^2=5 a^8 b^2 \times(-3) b^2$
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:
$16 m-4 m^2$
AnswerThe greatest common factor of the term
16 m and $4 m^2$ of the expression
Also, we can write $16 m-14 m .4$ and $4 m^2=4 m . m$
Therefore, $16 m-4 m^2=(4 m .4)-(4 m . m)$
$=4 m(4-m)$
View full question & answer→Question 622 Marks
Factorize of the following expressions:
$a(a+b-c)-b c$
Answer$a(a+b-c)-b c=a^2+a b-a c-b c$
$=\left(a^2-a c\right)+(a b-b c)$
$=a(a-c)+b(a-c)$
$=(a+b)(a-c)[$ taking $(a-c)$ as the common factor]
View full question & answer→Question 632 Marks
Factorize of the following algebraic expressions:
$9 a(6 a-5 b)-12 a^2(6 a-5 b)$
Answer$9 a(6 a-5 b)-12 a^2(6 a-5 b)$
$=\left(9 a-12 q a^2\right)(6 a-5 a)$ [taking $(6 a-5 b)$ as the common factor]
$=3 a(3-4)(6 a-5 b)\left[\right.$ taking $3 a$ as the common factor of the quadratic eqn. $\left.\left(9 a-12 a^2\right)\right]$
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:
$a(a-2 b-c)+2 b c$
Answer$a(a-2 b-c)+2 b c=a^2-2 a b-a c+2 b c$
$=\left(a^2-a c\right)+(2 b c-2 a b)$
$=a(a-c)+2 b(c-a)[\text { since, } 9 c-a)=-(a-c)]$
$=a(a-c)-2 b(a-c)$
$=(a-2 b)(a-c)[\text { taking }(a-c) \text { as the common factor] }$
View full question & answer→Question 672 Marks
Solve:
$(3-2 a)^2-25 a^2$
Answer$(3-2 a)^2-25 a^2$
$=(3+2 a)^2-25 a^2$
$=[(3 a+2 a)-5 a][(3+2 a)+5 a]$
$=(3+2 a-5 a)(3+2 a+5 a)$
$=(3-3 a)(3+7 a)$
$=3(1-a)(3+7 a)$
View full question & answer→Question 682 Marks
Factorize of the following expressions:
ab - a - b + 1
Answerab - 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:
$x^2+y-x y-x$
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$x^3-y^2+x-x^2 y^2$
$=\left(x^3+x\right)-\left(x^2 y^2+y^2\right) \\
=x\left(x^2+1\right)-y^2\left(x^2+1\right)$
$=\left(x-y^2\right)\left(x^2+1\right)\left[\text { taking }\left(x^2+1\right)\right. \text { as the common factor] }$
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
Answeraxy + 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:-
$-4 a^2+4 a b-4 c a$
AnswerThe greatest common factor of the term
$-4 a^2+4 a b$ and $4 c a$ of the expression
$-4 a^2+4 a b-4 c a$
Also, we can write $-4 a^2=(-4 a \cdot a), 4 a b=-4 a \cdot(-b)-4 c a$ and $4 c a=(-4 a \cdot c)$
Therefore, $-4 a^2+4 a b-4 c a=(-4 a \cdot a)+(-4 a \cdot(-b))-(4 a \cdot c)$
$-4 a(a-b)$
View full question & answer→Question 762 Marks
Factories:
$y^2+5 y-36$
AnswerTo factories $y^2+5 y-36$, we will find two number $p$ and $q$ such that $p+q=5$ and $p q=-36$
Now,
$9+(-4)=5$
And
$9 \times(-4)=-36$
Splittiong the middle term $5 y$ in the given quadratic as $-4 y+9 y$, we get:
$y^2+5 y-36=y^2-4 y+9 y-36$
$=(y 2-4 y)+(9 y-36)$
$=y(y-4)+9(y-4)$
$=(y+9)(y-4)$
View full question & answer→Question 772 Marks
Factorize:
$x^2 y z+x y^2 z+x y z^2$
AnswerThe greatest common factor of the term
$x^2 y z, x y^2 z$ and $x y z^2$ of the expression
$x^2 y z+x y^2 z+x y z^2$ is $x y z$.
Also, we can write $x^2 y z=(x y z . x),\left(x y^2 z=x y z . y\right), x y^2 z=(x y z . z)$
Therefore, $x^2 y z+x y^2 z+x y z^2=(x y z . x)+(x y z . y)+(x y z . z)$
View full question & answer→Question 782 Marks
Factorize of the following expressions:
$Lm ^2- mn ^2- Lm + n ^2$
Answer$Lm ^2- mn ^2- Lm + n ^2$
$=\left( Lm ^2- Lm \right)+\left( n ^2- mn ^2\right)$
$=\operatorname{Lm}(m-1)+ n ^2(1-m)$
$=L m(m-1)-n^2(m-1)[$ since, $(1-m)=-(m-1)]$
$=\left( Lm - n ^2\right)(m-1)[$ taking $(m-1)$ as the common factor]
View full question & answer→Question 792 Marks
Factorize of the following expressions:
$1+x+x y+x^2 y$
Answer$1+x+x y+x^2 y$
$=(1+x)+\left(x y+x^2 y\right)$
$=(1+x)+x y(1+x)$
$=(1+x y)(1+x)[\text { taking }(1+x) \text { as the common factor }]$
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 $a x^2 y=(x y . a x), b x y^2=(x y . b y), c x y z=(x y . c z)$
Therefore, $a x^2 y+b x y^2+c x y z=(x y \cdot a x)+(x y . b y)+(x y . c z)$
$=x y(a x+b y+c z)$
View full question & answer→Question 822 Marks
Solve:
$9(a-b)^2-100(x-y)^2$
Answer$9(a-b)^2-100(x-y)^2$
$=[3(a-b)]^2-[10(x-y)]^2$
$=[3(a-b)-10(x-y)][3(a-b)+10(x-y)]$
$=(3 a-3 b-10 x+10 y)(3 a-3 b+10 x-10)$
View full question & answer→Question 832 Marks
Factorize:
$5 x-15 x^2$
AnswerThe greatest common factor of the terms $5 x$ and $15 x^2$ of the expression $5 x-15 x^2$ is $5 x$
Now,
$5 x=5 x \cdot(-1) \text { and }-15 x^2=5 x \cdot(-3 x)$
Hence, the expression $5 x-15 x^2$ can be factorised as $5 x(1-3 x)$
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$144 a^2-289 b^2$
$= (12 a)^2-(17 b)^2$
$= (12 a-17 b)(12 a+17 b)$
View full question & answer→Question 862 Marks
Solve:
$a^2+2 a b+b^2-c^2$
Answer$a^2+2 a b+b^2-c^2$
$=\left(a^2+2 a b+b^2\right)-c^2$
$=\left(a^2+2 \times a \times b+b^2\right)-c^2$
$=(a+b)^2-c^2$
$=[(a+b)-c][(a+b)+c]$
$=(a+b-c)(a+b+c)$
View full question & answer→Question 872 Marks
Factorize of the following expressions:
2ax + bx + 2ay + by
Answer2ax + 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$18 a^2 x^2-32$
$=2\left(9 a^2 x^2-16\right)$
$=2\left[(3 a x)^2-4^2\right]$
$=2\left[(3 a x)^2-4^2\right]$
$=(3 a x-4)(3 a x+4)$
View full question & answer→Question 892 Marks
Factorize of the following expressions:
$x^2-2 a x-2 a b+b x$
Answer$x^2-2 a x-2 a b+b x$
$=\left(x^2-2 a x\right)+(b x-2 a b)$
$=x(x-2 a)+b(x-2 a)$
$=(x+b)(x-2 a)[\text { taking }(x-2 a) \text { as the common factor] }$
$=(x-2 a)(x+b)$
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
$20 x^3,-40 x^2$ and $80 x$ of the expression $20 x^3-40 x^2+80 x$ is $20 x$
Now,
$20 x^3=20 x \cdot x^2-40 x^2=20 x \cdot-2 x \text { and } 80 x=20 x \cdot 4$
Hence, the expression $20 x^3-40 x^2+80 x$ can be factorised as $20 x\left(x^2-2 x+4\right)$
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$a b\left(x^2+1\right)+x\left(a^2+b^2\right)=a b x^2+a b+a^2 x+b^2 x$
$=\left(a b x^2+a^2 x\right)+\left(b^2 x+a b\right)$
$=a x(b x+a)+b(b x+a)$
$=(a x+b)(b x+a)[$ taking $(b x+a)$ as the common facotor]
View full question & answer→Question 962 Marks
Factorize of the following algebraic expression:
$a(x-y)+2 b(y-x)+c(x-y)^2$
Answer$a(x-y)+2 b(y-x)+c(x-y)^2$
$=a(x-y)-2 b(x-y)+c(x-y)^2[(y-x)=-(x-y)$
$=[a-2 b+c(x-y)](x-y)$
$=(a-2 b+c x-c y)(x-y)$
View full question & answer→Question 972 Marks
Factorize of the following expressions:
$x^2+x y+x z+y z$
Answer$x^2+x y+x z+y z$
$=\left(x^2+x y\right)+(x z+y z)$
$=x(x+y)+z(x+y)$
$=(x+z)(x+y)[\text { taking }(x+y) \text { as the common factor] }$
$=(x+y)(x+z)$
View full question & answer→