Question 12 Marks
Factorise:
$8 x^2 y^3-x^5$
Answer$8 x^2 y^3-x^5$
$=x^2\left(8 y^3-x^3\right)$
$=x^2\left[(2 y)^3-x^3\right]$
$=x^2\left[(2 y-x)\left[(2 y)^2+(2 y)(x)+x^2\right]\right.$
$=x^2(2 y-x)\left(4 y^2+2 x y+x^2\right)$
View full question & answer→Question 22 Marks
Factorise:
$6 x^2+17 x+12$
Answer$6 x^2+17 x+12$
$=6 x^2+9 x+8 x+12$
$=3 x(2 x+3)+4(2 x+3)$
$=(2 x+3)(3 x+4)$
View full question & answer→Question 32 Marks
Factorise:$2\sqrt{3}\text{x}^2+\text{x}-5\sqrt{3}$
Answer$2\sqrt{3}\text{x}^2+\text{x}-5\sqrt{3}$$=2\sqrt{3}\text{x}^2+6\text{x}-5\text{x}-5\sqrt{3}$
$=2\sqrt{3}\text{x}\big(\text{x}+\sqrt{3}\big)-5\big(\text{x}+\sqrt{3}\Big)$
$=\big(\text{x}+\sqrt{3}\big)\big(2\sqrt{3}\text{x}-5\big)$
View full question & answer→Question 42 Marks
Factorise:
$2 x^2+3 x-90$
Answer$2 x^2+3 x-90$
$=2 x^2-12 x+15 x-90$
$=2 x(x-6)+15(x-6)$
$=(x-6)(2 x+15)$
View full question & answer→Question 52 Marks
Factorise:$\text{x}^2-2\sqrt{3}\text{x}-24$
Answer$\text{x}^2-2\sqrt{3}\text{x}-24$$=\text{x}^2-4\sqrt{3}\text{x}+2\sqrt{3}\text{x}-24$
$=\text{x}(\text{x}-4\sqrt{3})+2\sqrt{3}(\text{x}-4\sqrt{3})$
$=\text{x}(\text{x}-4\sqrt{3})(\text{x}+2\sqrt{3})$
View full question & answer→Question 62 Marks
Factorise:
$x^2-x-156$
Answer$x^2-x-156$
$=x^2-13 x+12 x-156$
$=x(x-13)+12(x-13)$
$=(x-13)(x+12)$
View full question & answer→Question 72 Marks
Factorise:
$a^2-b^2-4 a c+4 c^2$
Answer$a^2-4 a c+4 c^2-b^2$
$=a^2-4 a c+4 c^2-b^2$
$=a^2-2 \times a \times 2 c+(2 c)^2-b^2$
$=(a-2 c)^2-b^2\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=(a-2 c+b)(a-2 c-b)$
View full question & answer→Question 82 Marks
Factorise:
$x^2+7 x-98$
Answer$x^2+7 x-98$
$=x^2+14 x-7 x-98$
$=x(x+14)-7(x+14)$
$=(x+14)(x-7)$
View full question & answer→Question 92 Marks
Factorise:
$2 x^2+11 x-21$
Answer$2 x^2+11 x-21$
$=2 x^2+14 x-3 x-21$
$=2 x(x+7)-3(x+7)$
$=(x+7)(2 x-3)$
View full question & answer→Question 102 Marks
Factorise:
$(3 a-2 b)^3+(2 b-5 c)^3+(5 c-3 a)^3$
AnswerPut $(3 a-2 b)=x,(2 b-5 c)=y$ and $(5 c-3 a)=z$.
We have:
$x+y+z=3 a-2 b+2 b-5 c+5 c-3 a=0$
Now,
$(3 a-2 b)^3+(2 b-5 c)^3+(5 c-3 a)^3=x^3+y^3+z^3$
$=3 x y z\left[\text { Here, } x+y+z=0 . \text { So, } x^3+y^3+z^3\right]$
$=(3 a-2 b)(2 b-5 c)(5 c-3 a)$
View full question & answer→Question 112 Marks
Factorise:
$2 x^4-32$
Answer$2 x^4-32$
$=2\left(x^4-16\right)$
$=2\left[\left(x^2\right)^2-(4)^2\right]$
$=2\left[\left(x^2-4\right)\left(x^2+4\right)\right]$
$=2\left[\left(x^2-2^2\right)\left(x^2+4\right)\right]$
$=2\left[(x-2)(x+2)\left(x^2+4\right)\right]$
$=2(x-2)(x+2)\left(x^2+4\right)$
View full question & answer→Question 122 Marks
Factorise:
$25 x^2+4 y^2+9 z^2-20 x y-12 y z+30 x z$
AnswerWe have:
$25 x^2+4 y^2+9 z^2-20 x y-12 y z+30 x z$
$=(5 x)^2+(-2 y)^2+(3 z)^2+2(5 x)(-2 y)+2(-2 y)(3 z)+2(3 z)(5 x)$
$=[(5 x)+(-2 y)+(3 z)]^2$
$=(5 x-2 y+3 z)^2$
View full question & answer→Question 132 Marks
Expand:$\Big(3\text{a}+\frac{1}{4\text{b}}\Big)^3$
Answer$\Big(3\text{a}+\frac{1}{4\text{b}}\Big)^3$$=(3\text{a})^3+\Big(\frac{1}{4\text{b}}\Big)^3+3(3\text{a})^2\Big(\frac{1}{4\text{b}}\Big)+3(3\text{a})\Big(\frac{1}{4\text{b}}\Big)^2$
$=27\text{a}^3+\frac{1}{64\text{b}^3}+\frac{27\text{a}^2}{4\text{b}}+\frac{9\text{a}}{16\text{b}^2}$
View full question & answer→Question 142 Marks
Factorise:
$64 a^3-343$
Answer$64 a^3-343$
$=(4 a)^3-(7)^3$
$=(4 a-7)\left[(4 a)^2+(4 a)(7)+(7)^2\right]$
$=(4 a-7)\left(16 a^2+28 a+49\right)$
$=64+112+196 a-112-196 a-343$
$=64 a^3-343$
View full question & answer→Question 152 Marks
Factorise:
$18 x^2+3 x-10$
Answer$18 x^2+3 x-10$
$=18 x^2-12 x+15 x-10$
$=6 x(3 x-2)+5(3 x-2)$
$=(6 x+5)(3 x-2)$
View full question & answer→Question 162 Marks
Factorise:
$a^2-b^2+2 b c-c^2$
Answer$a^2-b^2+2 b c-c^2$
$=a^2-\left(b^2-2 b c+c^2\right)$
$=a^2-(b-c)^2$
$=[a-(b-c)][a+(b-c)]$
$=(a-b+c)(a+b-c)$
View full question & answer→Question 172 Marks
Factorise:
$9 x^2+16 y^2+4 z^2-24 x y+16 y z-12 x z$
AnswerWe have:
$9 x^2+16 y^2+4 z^2-24 x y+16 y z-12 x z$
$=(2 x)^2+(3 y)^2+(-4 z)^2+2(2 x)(3 y)+2(3 y)(-4 z)+2(-4 z)(2 x)$
$=[(2 x)+(3 y)+(-4 z)]^2$
$=(2 x+3 y-4 z)^2$
View full question & answer→Question 182 Marks
Factorise:
$x^2+18 x+32$
Answer$x^2+18 x+32$
$=x^2+16 x+2 x+32$
$=x(x+16)+2(x+16)$
$=(x+16)(x+2)$
View full question & answer→Question 192 Marks
Factorise:$\text{x}^2+7\sqrt{6}+60$
Answer$\text{x}^2+7\sqrt{6}+60$$=\text{x}^2+2\sqrt{6}\text{x}+5\sqrt{6}\text{x}+60$
$=\text{x}(\text{x}+2\sqrt{6})+5\sqrt{6}(\text{x}+2\sqrt{6})$
$=(\text{x}+2\sqrt{6})(\text{x}+5\sqrt{6})$
View full question & answer→Question 202 Marks
Factorise:$\frac{3}{5}\text{x}^2-\frac{19}{5}\text{x}+4$
Answer$\frac{3}{5}\text{x}^2-\frac{19}{5}\text{x}+4$$=\frac{3}{5}\text{x}^2+\frac{4}{5}\text{x}-3\text{x}+4$
$=\frac{\text{x}}{5}(3\text{x}-4)-1(3\text{x}-4)$
$=\Big(\frac{\text{x}}{5}-1\Big)(3\text{x}-4)$
View full question & answer→Question 212 Marks
Factorise:$\text{x}^2-2\sqrt{2}\text{x}-30$
Answer$\text{x}^2-2\sqrt{2}\text{x}-30$$=\text{x}^2-5\sqrt{2}\text{x}+3\sqrt{2}\text{x}-30$
$=\text{x}(\text{x}+5\sqrt{2})-3\sqrt{2}(\text{x}+5\sqrt{2})$
$=(\text{x}-5\sqrt{2})(\text{x}+3\sqrt{2})$
View full question & answer→Question 222 Marks
Factorise:
$2 a^3+16 b^3-5 a-10 b$
Answer$2 a^3+16 b^3-5 a-10 b$
$=2\left(a^3+8 b^3\right)-5(a+2 b)$
$=2\left[(a)^3+(2 b)^3\right]-5(a+2 b) \text { Since } a^3+b^3=\left(a+b\left(a^2-a \times b+b^2\right)\right.$
$=2(a+2 b)\left[(a)^2-a \times 2 b+(2 b)^2\right]-5(a+2 b)$
$=(a+2 b)\left[2\left(a^2-2 a b+4 b^2\right)-5\right]$
View full question & answer→Question 232 Marks
Evaluate:
$(995)^2$
Answer$(995)^2=(1000-5)^2$
$=[(1000)+(-5)]^2$
$=(1000)^2+2 \times(1000) \times(-5)+(-5)^2$
$=1000000-10000+25$
$=990025$
View full question & answer→Question 242 Marks
Factorise:
$40+3 x-x^2$
Answer$40+3 x-x^2$
$=40+8 x-5 x-x^2$
$=8(5+x)-x(5+x)$
$=(5+x)(8-x)$
View full question & answer→Question 252 Marks
Factorise:
$x^6-7 x^3-8$
AnswerGiven equation is $x^6-7 x^3-8$.
Putting $x^3=y$, we get
$y^2-7 y-8$
$=y^2-8 y+y-8$
$=y(y-8)+1(y-8)$
$=(y-8)(y+1)$
$=\left(x^3-8\right)\left(x^3+1\right)$
$=\left(x^3-2^3\right)\left(x^3+1^3\right)$
$=(x-2)\left(x^2+2 x+4\right)(x+1)\left(x^2-x+1\right)$
$=(x-2)(x+1)\left(x^2+2 x+4\right)\left(x^2-x+1\right)$
View full question & answer→Question 262 Marks
Factorise:$7\text{x}^2+2\sqrt{14}\text{x}+2$
Answer$7\text{x}^2+2\sqrt{14}\text{x}+2$$=7\text{x}^2+\sqrt{2}\big(\sqrt{7}\text{x}\big)+\sqrt{2}\big(\sqrt{7}\text{x}\big)+2$
$=\sqrt{7}\text{x}\big(\sqrt{7}\text{x}+\sqrt{2}\big)+\sqrt{2}\big(\sqrt{7}\text{x}+\sqrt{2}\big)$
$=\big(\sqrt{7}\text{x}+\sqrt{2}\big)\big(\sqrt{7}\text{x}+\sqrt{2}\big)=\big(\sqrt{7}\text{x}+\sqrt{2}\big)^2$
View full question & answer→Question 272 Marks
Factorise:$\frac{\text{x}^3}{216}-8\text{y}^3$
Answer$\frac{\text{x}^3}{216}-8\text{y}^3$$=\Big(\frac{\text{x}}{6}\Big)^3-(2\text{y})^3$
$=\Big(\frac{\text{x}}{6}-2\text{y}\Big)\bigg[\Big(\frac{\text{x}}{6}\Big)^2+\Big(\frac{\text{x}}{6}\Big)(2\text{y})+(2\text{y})^2\bigg]$
$=\Big(\frac{\text{x}}{6}-2\text{y}\Big)\Big(\frac{\text{x}^2}{36}+\frac{\text{xy}}{3}+4\text{y}^2\Big)$
View full question & answer→Question 282 Marks
Factorise:$\sqrt{2}\text{x}^2+3\text{x}+\sqrt{2}$
Answer$\sqrt{2}\text{x}^2+3\text{x}+\sqrt{2}$$=\sqrt{2}\text{x}^2+\text{x}+2\text{x}+\sqrt{2}$
$=\text{x}\big(\sqrt{2}\text{x}+1\big)+\sqrt{2}\big(\sqrt{2}\text{x}+1\big)$
$=\big(\sqrt{2}\text{x}+1\big)\big(\text{x}+\sqrt{2}\big)$
View full question & answer→Question 292 Marks
Factorise:
$9 x^2+18 x+8$
Answer$9 x^2+18 x+8$
$=9 x^2+12 x+6 x+8$
$=3 x(3 x+4)+2(3 x+4)$
$=(3 x+4)(3 x+2)$
View full question & answer→Question 302 Marks
Factorise:
$6 x^2-11 x-35$
Answer$6 x^2-11 x-35$
$=6 x^2-21 x+10 x-35$
$=3 x(2 x-7)+5(2 x-7)$
$=(2 x-7)(3 x+5)$
View full question & answer→Question 312 Marks
Factorise:
$15x^2 + 2x - 8$
Answer$15x^2 + 2x - 8$
$= 15x^2 - 10x + 12x - 8$
$= 5x(3x - 2) + 4(3x - 2)$
$= (3x - 2)(5x + 4)$
View full question & answer→Question 322 Marks
Factorise:
$(a+b)^3-(a-b)^3$
AnswerWe know that, Since $a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
Therefore,
$(a+b)^3-(a-b)^3$
$=[a+b-(a-b)]\left[(a+b)^2+(a+b)(a-b)+(a-b)^2\right.$
$=(a+b-a+b)\left[a^2+b^2+2 a b+a^2-b^2+a^2+b^2-2 a b\right]$
$=2 b\left(3 a^2+b^2\right)$
View full question & answer→Question 332 Marks
Factorise:
$8(x+y)^3-27(x-y)^3$
Answer$8(x+y)^3-27(x-y)^3$
$\left.=\left[2^3(x+y)^3\right]-\left[3^3(x-y)\right)^3\right]$
$\left.=[2(x+y)-3(x-y)\}[2(x+y)]^2+2(x+y) 3(x-y)+[3(x-y)]^2\right\}$
$=(2 x+2 y-3 x+3 y)\left\{\left[4\left(x^2+y^2+2 x y\right)\right]+6\left(x^2-y^2\right)+\left[9\left(x^2+y^2-2 x y\right]\right\}\right.$
$=(-x+5 y)\left\{4 x^2+4 y^2+8 x y+6 x^2-6 y^2+9 x^2+9 y^2-18 x y\right\}$
$=(-x+5 y)\left(19 x^2+7 y^2-10 x y\right)$
View full question & answer→Question 342 Marks
Factorise:
$(2 a+1)^3+(a-1)^3$
Answer$(2 a+1)^3+(a-1)^3$
$=(2 a+1+a-1)\left[(2 a+1)^2-(2 a+1)(a-1)+(a-1)^2\right]$
$=(3 a)\left[4 a^2+4 a+1-2 a^2+2 a-a+1+a^2-2 a+1\right]$
$=3 a\left(3 a^2+3 a+3\right)$
$=9 a\left(a^2+a+1\right)$
View full question & answer→Question 352 Marks
Factorise:
$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$
$=a b x^2+a^2 x+a b+b^2 x$
$=a x(b x+a)+b(b x+a)$
$=(b x+a)(a x+b)$
View full question & answer→Question 362 Marks
Factorise:$\text{x}^2+\frac{1}{\text{x}^2}-2-3\text{x}+\frac{3}{\text{x}}$
Answer$\text{x}^2+\frac{1}{\text{x}^2}-2-3\text{x}+\frac{3}{\text{x}}$$=\Big(\text{x}-\frac{1}{\text{x}}\Big)^2-3\Big(\text{x}-\frac{1}{\text{x}}\Big)$
$=\Big(\text{x}-\frac{1}{\text{x}}\Big)\Big(\text{x}-\frac{1}{\text{x}}-3\Big)$
View full question & answer→Question 372 Marks
Factorise:$\text{x}^2-3\sqrt{5}\text{x}-20$
Answer$\text{x}^2-3\sqrt{5}\text{x}-20$$=\text{x}^2-4\sqrt{5}\text{x}+\sqrt{5}\text{x}-20$
$=\text{x}(\text{x}-4\sqrt{5})+\sqrt{5}(\text{x}-4\sqrt{5})$
$=\text{x}(\text{x}-4\sqrt{5})(\text{x}+\sqrt{5})$
View full question & answer→Question 382 Marks
Factorise:
$1-27 a^3$
Answer$1-27 a^3$
$=(1)^3-(3 a)^3$
$=(1-3 a)\left[(1)^2+1 \times 3 a+(3 a)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=(1-3 a)\left(1+3 a+9 a^2\right)$^2)
View full question & answer→Question 392 Marks
Expand:
$(2 a+5 b+7 c)^2$
Answer$(2 a+5 b+7 c)^2$
$=[(2 a)+(-5 b)+(-7 c)]^2$
$=(2 a)^2+(-5 b)^2+(-7 c)^2+2(2 a)(-5 b)+2(-5 b)(-7 c)+2(2 a)(-7 c)$
$=4 a^2+25 b^2+49 c^2-20 a b+70 b c-28 a c$
View full question & answer→Question 402 Marks
Factorise:$2\text{x}^2-\text{x}+\frac{1}{8}$
Answer$2\text{x}^2-\text{x}+\frac{1}{8}$$=2\text{x}^2-\frac{1}{2}\text{x}-\frac{1}{2}\text{x}+\frac{1}{8}$
$=\frac{\text{x}}{2}(4\text{x}-1)-\frac{1}{8}(4\text{x}-1)$
$=\Big(\frac{\text{x}}{2}-\frac{1}{8}\Big)(4\text{x}-1)$
View full question & answer→Question 412 Marks
Factorise:
$a b\left(x^2+y^2\right)-x y\left(a^2+b^2\right)$
Answer$a b\left(x^2+y^2\right)-x y\left(a^2+b^2\right)$
$=a b x^2+a b y^2-a^2 x y-b^2 x y$
$=a b x^2-a^2 x y+a b y^2-b^2 x y$
$=a x(b x-a y)+b y(a y-b x)$
$=(b x-a y)(a x-b y)$
View full question & answer→Question 422 Marks
Factorise:
$6-x-x^2$
Answer$6-x-x^2$
$=6+2 x-3 x-x^2$
$=2(3+x)-x(3+x)$
$=(3+x)(2-x)$
View full question & answer→Question 432 Marks
Factorise:
$x^2-32 x-105$
Answer$x^2-32 x-105$
$=x^2-35 x+3 x-105$
$=x(x-35)+3(x-35)$
$=(x-35)(x+3)$
View full question & answer→Question 442 Marks
Factorise:$\text{x}^2+6\sqrt{6}\text{x}+48$
Answer$\text{x}^2+6\sqrt{6}\text{x}+48$$=\text{x}^2+4\sqrt{6}\text{x}+2\sqrt{6}\text{x}+48$
$=\text{x}(\text{x}+4\sqrt{6})+2\sqrt{6}(\text{x}+4\sqrt{6})$
$=(\text{x}+4\sqrt{6})(\text{x}+2\sqrt{6})$
View full question & answer→Question 452 Marks
Factorise:$\text{x}^2-2+\frac{1}{\text{x}^2}-\text{y}^2$
Answer$\text{x}^2-2+\frac{1}{\text{x}^2}-\text{y}^2$$=\Big(\text{x}^2-2(\text{x}^2)\Big(\frac{1}{\text{x}^2}\Big)+\frac{1}{\text{x}^2}\Big)-\text{y}^2$
$=\Big(\text{x}-\frac{1}{\text{x}}\Big)^2-\text{y}^2$
$=\Big(\text{x}-\frac{1}{\text{x}}+\text{y}\Big)\Big(\text{x}-\frac{1}{\text{x}}-\text{y}\Big)$
View full question & answer→Question 462 Marks
Factorise:
$8(3 a-2 b)^2-10(3 a-2 b)$
Answer$8(3 a-2 b)^2-10(3 a-2 b)$
$=(3 a-2 b)[8(3 a-2 b)-10]$
$=(3 a-2 b) 2[4(3 a-2 b)-5]$
$=2(3 a-2 b)(12 a-8 b-5)$
View full question & answer→Question 472 Marks
Factorise:
$7 a^3+56 b^3$
Answer$7 a^3+56 b^3$
$=7\left(a^3+8 b^3\right)$
$=7\left[(a)^3+(2 b)^3\right]$
$=7(a+2 b)\left[a^2-a \times 2 b+(2 b)^2\right] \text { Since } a^3+b^3=(a+b)\left(a^2-a \times b+b^2\right)$
$=7(a+2 b)\left(a^2-2 a b+4 b^2\right)$
View full question & answer→Question 482 Marks
Factorise:
$125 a^3+b^3+64 c^3-60 a b c$
Answer$125 a^3+b^3+64 c^3-60 a b c$
$=(5 a)^3+(b)^3+(4 c)^3-3 \times 5 a \times b \times 4 c$
$=(5 a+b+4 c)\left[(5 a)^2+(b)^2+(4 c)^2-5 a \times b-b \times 4 c-5 a \times 4 c\right]$
$=(5 a+b+4 c)\left(25 a^2+b^2+16 c^2-5 a b-4 b c-20 a c\right.$
View full question & answer→Question 492 Marks
Expand:$\Big(1+\frac{2}{3}{\text{a}}\Big)^3$
Answer$\Big(1+\frac{2}{3}{\text{a}}\Big)^3$$=\Big(\frac{2}{3}\text{a}\Big)^3+3\times\Big(\frac{2}{3}\text{a}\Big)^2\times1+3\text{a}\frac{2}{3}\text{a}\times(1)^2+(1)^3$
$=\frac{8}{27}\text{a}^3+\frac{4}{3}\text{a}^2+2\text{a}+1$
View full question & answer→Question 502 Marks
Factorise:
$3 a^3 b-243 a b^3$
Answer$3 a^3 b-243 a b^3$
$=3 a b\left(a^2-81 b^2\right)$
$=3 a b\left[(a)^2-(9 b)^2\right]$
$=3 a b(a+9 b)(a-9 b)\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
View full question & answer→Question 512 Marks
If a, b, c are all nonzero and a + b + c = 0, prove that:$\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}=3$
Answer$a + b + c = 0 \Rightarrow a^3 + b^3 + c^3 = 3abc$
Thus, We have:$\Big(\frac{\text{a}^2}{\text{bc}}+\frac{\text{b}^2}{\text{ca}}+\frac{\text{c}^2}{\text{ab}}\Big)=\frac{\text{a}^3+\text{b}^3+\text{c}^3}{\text{abc}}$
$=\frac{3\text{abc}}{\text{abc}}$
$=3$
View full question & answer→Question 522 Marks
Factorise:$2\text{x}^2+3\sqrt{3}\text{x}+3$
Answer$2\text{x}^2+3\sqrt{3}\text{x}+3$$2\text{x}^2+2\sqrt{3}\text{x}+\sqrt{3}\text{x}+3$
$=2\text{x}\big(\text{x}+\sqrt{3}\big)+\sqrt{3}\big(\text{x}+\sqrt{3}\big)$
$=\big(\text{x}+\sqrt{3}\big)\big(2\text{x}+\sqrt{3}\big)$
View full question & answer→Question 532 Marks
Expand:$\Big(3\text{x}-\frac{5}{\text{x}}\Big)^3$
Answer$\Big(3\text{x}-\frac{5}{\text{x}}\Big)^3$$=(3\text{x})^3-\Big(\frac{5}{\text{x}}\Big)^3-3(3\text{x})^2\Big(\frac{5}{\text{x}}\Big)+3(3\text{x})\Big(\frac{5}{\text{x}}\Big)^2$
$=27\text{x}^3-\frac{125}{\text{x}^3}-135\text{x}+\frac{225}{\text{x}}$
View full question & answer→Question 542 Marks
Evaluate:
$(107)^2$
Answer$(107)^2=(100+7)^2$
$=(100)^2+2 \times(100) \times(7)+(7)^2$
$=10000+1400+49$
$=11449$
View full question & answer→Question 552 Marks
Expand:$(5a - 3b)$ ${ }^3$
Answer$(5 a-3 b)^3$
$=(5 a)^3-(3 b)^3-3(5 a)^2(3 b)+3(5 a)(3 b)^2$
$=125 a^3-27 b^3-225 a^2 b+135 a b^2$
View full question & answer→Question 562 Marks
Factorise:
$x^2-32 x-105$
Answer$x^2-32 x-105$
$=x^2-35 x+3 x-105$
$=x(x-35)+3(x-35)$
$=x(x-35)(x+3)$
View full question & answer→Question 572 Marks
Factorise:$6\sqrt{3}\text{x}^2-47\text{x}+5\sqrt{3}$
Answer$6\sqrt{3}\text{x}^2-47\text{x}+5\sqrt{3}$$=6\sqrt{3}\text{x}^2-45\text{x}-2\text{x}+5\sqrt{3}$
$=3\sqrt{3}\text{x}\big(2\text{x}-5\sqrt{3}\big)-1\big(2\text{x}-5\sqrt{3}\big)$
$=\big(2\text{x}-5\sqrt{3}\big)\big(3\sqrt{3}\text{x}-1\big)$
View full question & answer→Question 582 Marks
Factorise:
$(a+b)^3-a-b$
Answer$(a+b)^3-a-b$
$=(a+b)^3-(a+b)$
$=(a+b)\left[(a+b)^2-1^2\right]\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=(a+b)(a+b+1)(a+b-1)$
View full question & answer→Question 592 Marks
Factorise:$2\sqrt{2}\text{a}^3+16\sqrt{2}\text{b}^3+\text{c}^3-12\text{abc}$
Answer$x =2\sqrt{2}\text{a}^3+16\sqrt{2}\text{b}^3+\text{c}^3-12\text{abc}$$=\big(\sqrt{2}\text{a}\big)^3+\big(2\sqrt{2}\text{b}\big)^3+\text{c}^3-3\times\big(\sqrt{2}\text{a}\big)\times\big(2\sqrt{2}\text{b}\big)\times(\text{c})$
$=\big(\sqrt{2}\text{a}+2\sqrt{2}\text{b}+\text{c}\big)\Big[\big(\sqrt{2}\text{a}\big)^2+(2\sqrt{2}\text{b})^2+\text{c}^2\\-\big(\sqrt{2}\text{a}\big)\times\big(2\sqrt{2}\text{b}\big)-\big(2\sqrt{2}\text{b}\big)\times(\text{c})-\big(\sqrt{2}\text{a}\big)\times(\text{c})\Big]$
$=\big(\sqrt{2}\text{a}+2\sqrt{2}\text{b}+\text{c}\big)\big(2\text{a}^2+8\text{b}^2+\text{c}^2-4\text{ab}-2\sqrt{2\text{bc}}-\sqrt{2}\text{ac}\big)$
View full question & answer→Question 602 Marks
Factorise:$\text{x}^2+\sqrt{2}\text{x}-24$
Answer$\text{x}^2+\sqrt{2}\text{x}-24$$=\text{x}^2+4\sqrt{2}\text{x}-3\sqrt{2}\text{x}-24$
$=\text{x}(\text{x}+4\sqrt{2})-3\sqrt{2}(\text{x}+4\sqrt{2})$
$=(\text{x}-4\sqrt{2})(\text{x}-3\sqrt{2})$
View full question & answer→Question 612 Marks
Factorise:
$10 x^2-9 x-7$
Answer$10 x^2-9 x-7$
$=10 x^2+5 x-14 x-7$
$=5 x(2 x+1)-7(2 x+1)$
$=(2 x+1)(5 x-7)$
View full question & answer→Question 622 Marks
Factorise:
$x^2-21 x+90$
Answer$x^2-21 x+90$
$=x^2-6 x-15 x+90$
$=x(x-6)-15(x-6)$
$=(x-6)(x-15)$
View full question & answer→Question 632 Marks
Factorise:
$1029-3 x^3$
Answer$1029-3 x^3$
$=3\left(343-x^3\right)$
$=3\left[(7)^3-x^3\right]$
$=3\left[(7-x)\left(7^2+7 x+x^2\right)\right]$
$=3(7-x)\left(49+7 x+x^2\right)$
View full question & answer→Question 642 Marks
Factorise:
$x^2+11 x+30$
Answer$x^2+11 x+30$
$=x^2+6 x+5 x+30$
$=x(x+6)+5(x+6)$
$=(x+6)(x+5)$
View full question & answer→Question 652 Marks
Factorise:
$25 x^2-10 x+1-36 y^2$
Answer$25 x^2-10 x+1-36 y^2$
$=\left(25 x^2-10 x+1\right)-36 y^2$
$=\left[(5 x)^2-2(5 x)(1)+(1)^2\right]-(6 y)^2$
$=(5 x-1)^2-(6 y)^2$
$=(5 x-1-6 y)(5 x-1+6 y)$
View full question & answer→Question 662 Marks
Factorise:$\frac{3}{2}\text{x}^2+16\text{x}+10$
Answer$\frac{3}{2}\text{x}^2+16\text{x}+10$$=\frac{3}{2}\text{x}^2+\text{x}+15\text{x}+10$
$=\frac{\text{x}}{2}(3\text{x}+2)+5(3\text{x}+2)$
$(3\text{x}+2)\Big(\frac{\text{x}}{2}+5\Big)$
View full question & answer→Question 672 Marks
Factorise:
$x-8 x y^3$
Answer$x-8 x y^3$
$=x\left(1-8 y^3\right)$
$=x\left[(1)^3-(2 y)^3\right]$
$=x(1-2 y)\left[(1)^2+1 \times 2 y+(2 y)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=x(1-2 y)\left(1+2 y+4 y^2\right)$
View full question & answer→Question 682 Marks
Factorise:$\text{x}^4+\frac{4}{\text{x}^4}$
Answer$\text{x}^4+\frac{4}{\text{x}^4}$$=\text{x}^4+\frac{4}{\text{x}^4}+4-4$
$=\big(\text{x}^2\big)^2+\Big(\frac{2}{\text{x}^2}\Big)^2+2\big(\text{x}^2\big)\Big(\frac{2}{\text{x}^2}\Big)-2^2$
$=\bigg[\big(\text{x}^2\big)^2+\Big(\frac{2}{\text{x}^2}\Big)^2+2\big(\text{x}^2\big)\Big(\frac{2}{\text{x}^2}\Big)\bigg]-2^2$
$=\Big[\text{x}^2+\frac{2}{\text{x}^2}\Big]^2-2^2$
$=\Big(\text{x}^2+\frac{2}{\text{x}^2}+2\Big)\Big(\text{x}^2+\frac{2}{\text{x}^2}-2\Big)$
View full question & answer→Question 692 Marks
Factorise:
$x^6-729$
Answer$x^6-729$
$=\left(x^2\right)^3-(9)^3$
$=\left(x^2-9\right)\left[\left(x^2\right)^2+x^2 \times 9+(9)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=\left(x^2-9\right)\left(x^4+9 x^2+81\right)$
$=(x+3)(x-3)\left[\left(x^2+9\right)^2-(3 x)^2\right]$
$=(x+3)(x-3)\left(x^2+3 x+9\right)\left(x^2-3 x+9\right)$
View full question & answer→Question 702 Marks
Factorise:
$27 a^3-b^3+8 c^3-18 a b c$
Answer$27 a^3-b^3+8 c^3-18 a b c$
$=(3 a)^3+(-b)^3+(2 c)^3-3 \times(3 a) \times(-b) \times(2 c)$
$=[3 a+(-b)+2 c]\left[(3 a)^2+(-b)^2+(2 c)^2-3 a(-b) 2 c-3 a \times 2 c\right]$
$=(3 a-b+2 c)\left(9 a^2+b^2+4 c^2+3 a b+2 b c-6 a c\right)$
View full question & answer→Question 712 Marks
Factorise:$1+\frac{27}{125}\text{a}^3+\frac{9\text{a}}{5}+\frac{27\text{a}^2}{25}$
Answer$1+\frac{27}{125}\text{a}^3+\frac{9\text{a}}{5}+\frac{27\text{a}^2}{25}$$=(1)^3+\Big(\frac{3}{5}\text{a}\Big)^3+3(1)^2\Big(\frac{3}{5}\text{a}\Big)+3(1)\Big(\frac{3}{5}\text{a}\Big)^2$
$=\Big(1+\frac{3}{5}\text{a}\Big)^3$
Hence, factorisation of $1+\frac{27}{125}\text{a}^3+\frac{9\text{a}}{5}+\frac{27\text{a}^2}{25}$ is $=\Big(1+\frac{3}{5}\text{a}\Big)^3$
View full question & answer→Question 722 Marks
Factorise:$21\text{x}^2-2\text{x}+\frac{1}{21}$
Answer$21\text{x}^2-2\text{x}+\frac{1}{21}$$21\text{x}^2-\text{x}-\text{x}+\frac{1}{21}$
$=21\text{x}\Big(\text{x}-\frac{1}{21}\Big)-1\Big(\text{x}-\frac{1}{21}\Big)$
$=\Big(\text{x}-\frac{1}{21}\Big)(21\text{x}-1)$
View full question & answer→Question 732 Marks
Factorise:
$a^3+0.008$
Answer$a^3+0.008$
$=(a)^3+(0.2)^3$
$=(a+0.2)\left[(a)^2-a \times 0.2+(0.2)^2\right] \text { Since } a^3+b^3=(a+b)\left(a^2-a \times b+b^2\right)$
$=(a+0.2)\left(a^2-0.2 a+0.04\right)$
View full question & answer→Question 742 Marks
Expand:
$(2b - b + c)^2$
Answer$(2b - b + c)^2 = [(2a) + (-b) + (c)]^2$
$= (2a)^2 + (-b)^2 + (c)^2 + 2(2a)(-b) + 2(-b)(c) + 4(a)(c)$
$= 14a^2 + b^2 + c^2 - 4ab - 2bc + 4ac$
View full question & answer→Question 752 Marks
Factorise:$\sqrt{3}\text{x}^2-10\text{x}+8\sqrt{3}$
Answer$\sqrt{3}\text{x}^2-10\text{x}+8\sqrt{3}$$=\sqrt{3}\text{x}^2+4\text{x}+6\text{x}+8\sqrt{3}$
$=\text{x}\big(\sqrt{3}\text{x}+4\big)+2\sqrt{3}\big(\sqrt{3}\text{x}+4\big)$
$=\big(\sqrt{3}\text{x}+4\big)\big(\text{x}+2\sqrt{3}\big)$
View full question & answer→Question 762 Marks
Factorise:$\text{x}^2+5\sqrt{5}\text{x}+30$
Answer$\text{x}^2+5\sqrt{5}\text{x}+30$$=\text{x}^2+2\sqrt{5}\text{x}+3\sqrt{5}\text{x}+30$
$=\text{x}(\text{x}+2\sqrt{5})+3\sqrt{5}(\text{x}+2\sqrt{5})$
$=(\text{x}+2\sqrt{5})(\text{x}+3\sqrt{5})$
View full question & answer→Question 772 Marks
Factorise:
$125 - 8x^3 - 27y^3 - 90xy$
Answer$125 - 8x^3 - 27y^3 - 90xy$
$= 5^3 + (-2x)^3 + (-3y)^3 - 3 \times 5 \times (-2x) \times (-3y)$
$= [5 + (-2x) + (-3y)][5^2 + (-2x)^2 + (-3y)^2 - 5 \times (-2x) - (-2x)(-3y) - 5 \times (-3y)]$
$= (5 - 2x - 3y)(25 + 4x^2 + 9y^2 + 10x - 6xy + 15y)$
View full question & answer→Question 782 Marks
Factorise:
$4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz$
AnswerWe have:
$4 x^2+9 y^2+16 z^2+12 x y-24 y z-16 x z$
$=(2 x)^2+(3 y)^2+(-4 z)^2+2(2 x)(3 y)+2(3 y)(-4 z)+2(-4 z)(2 x)$
$=[(2 x)+(3 y)+(-4 z)]^2$
$=(2 x+3 y-4 z)^2$
View full question & answer→Question 792 Marks
Find the product.
$(x + y - z)(x^2 + y^2 + z^2 - xy + yz + zx)$
Answer$(x + y - z)(x^2 + y^2 + z^2 - xy + yz + zx)$
$= [x + y + (-z)][x^2 + y^2 + (-z)^2 - xy - y \times (-z) - [-z] \times x]$
$= x^3 + y^3 + (-z)^3 - 3x \times y \times (-z)$
$= x^3 + y^3 + (-z)^3 - 3x \times y \times (-z)$
$= x^3 + y^3 - z^3 + 3xyz$
View full question & answer→Question 802 Marks
Factorise:
$x^2 - (a + b)x + ab$
Answer$x^2 - (a + b)x + ab$
$= x^2 - ax - bx + ab$
$= x(x - a) - b(x - a)$
$= (x - a)(x - b)$
View full question & answer→Question 812 Marks
Factorise:
$x^2 - 4x + 3$
Answer$x^2 - 4x + 3$
$= x^2 - 3x - x + 3$
$= x(x - 3) - 1(x - 3)$
$= (x - 3)(x - 1)$
View full question & answer→Question 822 Marks
Factorise:$\frac{2}{3}\text{x}^2-\frac{17}{3}\text{x}-28$
Answer$\frac{2}{3}\text{x}^2-\frac{17}{3}\text{x}-28$$=\frac{2}{3}\text{x}^2+\frac{7}{3}\text{x}-8\text{x}-28$
$=\frac{\text{x}}{2}(2\text{x}+7)-4(2\text{x}+7)$
$=\Big(\frac{\text{x}}{3}-4\Big)(2\text{x}+7)$
View full question & answer→Question 832 Marks
Factorise:$125\text{a}^3+\frac{1}{8}$
Answer$125\text{a}^3+\frac{1}{8}$We know that:
Since $a^2 + b^3 = (a + b)(a^2 - a \times b + b^2)$
Let us rewrite
$125\text{a}^3+\frac{1}{8}$
$=(5\text{a})^3+\Big(\frac{1}{2}\Big)^3$
$=\Big(5\text{a}+\frac{1}{2}\Big)\bigg[(5\text{a})^2-5\text{a}\times\frac{1}{2}+\Big(\frac{1}{2}\Big)^2\bigg]$
$=\Big(5\text{a}+\frac{1}{2}\Big)\Big(25\text{a}^2-\frac{5\text{a}}{2}+\frac{1}{4}\Big)$
View full question & answer→Question 842 Marks
Factorise:
$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[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=\left(x^2+25\right)(x+5)(x-5)$
View full question & answer→Question 852 Marks
Factorise:
$9 a^2+3 a-8 b-64 b^2$
Answer$9 a^2+3 a-8 b-64 b^2$
$=9 a^2-64 b^2+3 a-8 b$
$=(3 a)^2-(8 b)^2+(3 a-8 b)\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=(3 a+8 b)(3 a-8 b)+(3 a-8 b)$
$=(3 a-8 b)(3 a+8 b+1)$
View full question & answer→Question 862 Marks
Factorise:
$9-a^2+2 a b-b^2$
Answer$9-a^2+2 a b-b^2$
$=9-\left(a^2-2 a b+b^2\right)$
$=3^2-(a-b)^2\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=(3+a-b)(3-a+b)$
View full question & answer→Question 872 Marks
Expand:
$(-3 a+4 b-5 c)^2$
Answer$(-3 a+4 b-5 c)^2$
$=[(-3 a)+(4 b)+(-5 c)]^2$
$=(-3 a)^2+(4 b)^2+(-5 c)^2+2(-3 a)(4 b)+2(4 b)(-5 c)+2(-3 a)(-5 c)$
$=9 a^2+16 b^2+25 c^2-24 a b-40 b c+30 a c$
View full question & answer→Question 882 Marks
Factorise:
$a^{12}-b^{12}$
Answer$a^{12}-b^{12}$
$=\left(a^6\right)^2-\left(b^6\right)^2$
$=\left(a^6-b^6\right)\left(a^6+b^6\right)$
$=\left[\left(a^3\right)^2-\left(b^3\right)^2\right]\left[\left(a^2\right)^3+\left(b^2\right)^3\right]$
$=\left(a^3-b^3\right)\left(a^3+b^3\right)\left[\left(a^2+b^2\right)\left(a^4-a^2 b^2+b^4\right)\right]$
$=(a-b)\left(a^2+a b+b^2\right)(a+b)\left(a^2-a b+b^2\right)\left(a^2+b^2\right)\left(a^4-a^2 b^2+b^4\right)$
$=(a-b)(a+b)\left(a^2+b^2\right)\left(a^2+a b+b^2\right)\left(a^2-a b+b^2\right)\left(a^4-a^2 b^2+b^4\right)$
View full question & answer→Question 892 Marks
Evaluate:
$(99)^3$
Answer$(99)^3$
$=(100-1)^3$
$=(100)^3-(1)^3-3(100)^2 \times(1)+3(100)(1)^2$
$=1000000-1-30000+300$
$=1000300-30001$
$=970299$
View full question & answer→Question 902 Marks
Factorise:
$24 x^2-41 x+12$
Answer$24 x^2-41 x+12$
$=24 x^2-32 x-9 x+12$
$=8 x(3 x-4)-3(3 x-4)$
$=(3 x-4)(8 x-3)$
View full question & answer→Question 912 Marks
Factorise:
$27 x^3-y^3-z^3-9 x y z$
Answer$27 x^3-y^3-z^3-9 x y z$
$=(3 x)^3-y^3-z^3-3 \times(3 x) \times(-y) \times(-z)$
We know,
$a^3+b^3+c^3-3 a b c$
$=(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)$
$a=3 x, b=-y, c=-z$
$(3 x)^3-y^3-z^3-3 \times(3 x) \times(-y) \times(-z)$
$=(3 x-y-z)\left(9 x^2+y^2+z^2+3 x y-y z+3 x z\right)$
View full question & answer→Question 922 Marks
Evaluate:
$(103)^3$
Answer$(103)^3$
$=(100+3)^3$
$=(100)^3+(3)^3+3(100)^2 \times(3)+3(100)(3)^2$
$=1000000+27+90000+2700$
$=1092727$
View full question & answer→Question 932 Marks
Factorise:
$x^9-y^9$
Answer$x^9-y^9$
$=\left(x^3\right)^3-\left(y^3\right)^3$
$=\left[\left(x^3-y^3\right)\right]\left[\left(x^3\right)^2+x^3 y^3+\left(y^3\right)^2\right]$
$=\left[(x-y)\left(x^2+x y+y^2\right)\left(x^6+x^3 y^3+y^6\right)\right.$
View full question & answer→Question 942 Marks
Factorise:
$(a-b)^3+(b-c)^3+(c-a)^3$
Answer$(a-b)^3+(b-c)^3+(c-a)^3$
Putting $(a-b)=x,(b-c)=y \text { and }(c-a)=z$
We get: $(a-b)^3+(b-c)^3+(c-a)^3$
$=x^3+y^3+z^3[(x+y+z)=(a-b)+(b-c)+(c-a)=0]$
$=3 x y z\left[(x+y+z)=0 \Rightarrow x^3+y^3+z^3=3 x y z\right]$
$=3(a-b)(b-c)(c-a)$
View full question & answer→Question 952 Marks
Factorise:
$(a x+b y)^2+(b x-a y)^2$
Answer$(a x+b y)^2+(b x-a y)^2$
$=a^2 x^2+b^2 y^2+2 a b x y+b^2 x^2+a^2 y^2-2 a b x y$
$=a^2 x^2+b^2 y^2+b^2 x^2+a^2 y^2$
$=a^2 x^2+b^2 x^2+b^2 y^2+a^2 y^2$
$=x^2\left(a^2+b^2\right)+y^2\left(a^2+b^2\right)$
$=\left(a^2+b^2\right)\left(x^2+y^2\right)$
View full question & answer→Question 962 Marks
Factorise:$\text{x}^2+3\sqrt{3}\text{x}+6$
Answer$\text{x}^2+3\sqrt{3}\text{x}+6$$=\text{x}^2+2\sqrt{3}\text{x}+\sqrt{3}\text{x}+6$
$=\text{x}(\text{x}+2\sqrt{3})+\sqrt{3}(\text{x}+2\sqrt{3})$
$=(\text{x}+2\sqrt{3})(\text{x}+\sqrt{3})$
View full question & answer→Question 972 Marks
Factorise:
$a^3-0.064$
Answer$a^3-0.064$
$=(a)^3-(0.4)^3$
$=(a-0.4)\left[(a)^2+a \times 0.4+(0.4)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=(a-0.4)\left(a^2+0.4 a+0.16\right)$
View full question & answer→Question 982 Marks
Factorise:
$81 x^4-y^4$
Answer$81 x^4-y^4$
$=\left(9 x^2\right)^2-\left(y^2\right)^2$
$=\left(9 x^2-y^2\right)\left(9 x^2+y^2\right)$
$=\left[(3 x)^2-y^2\right]\left(9 x^2+y^2\right)$
$=(3 x-y)(3 x+y)\left(9 x^2+y^2\right)$
View full question & answer→Question 992 Marks
Factorise:
$8 a^3+125 b^3-64 c^3+120 a b c$
Answer$8 a^3+125 b^3-64 c^3+120 a b c$
$=(2 a)^3+(5 b)^3+(-4 c)^3-3 \times(2 a) \times(5 b) \times(-4 c)$
$=(2 a+5 b-4 c)\left[(2 a)^2+(5 b)^2+(-4 c)^2-(2 a)(5 b)-(5 b)(-4 c)-(2 a) \times(-4 c)\right]$
$=(2 a+5 b-4 c)\left(4 a^2+25 b^2+16 c^2-10 a b+20 b c+8 a c\right)$
View full question & answer→Question 1002 Marks
Factorise:
$(x+2)^3+(x-2)^3$
Answer$(x+2)^3+(x-2)^3$
$=[(x+2)+(x-2)]\left[(x+2)^2-(x+2)(x-2)+(x-2)^2\right]$
$=(2 x)\left(x^2+4 x+4-x^2+4+x^2-4 x+4\right)$
$=2 x\left(x^2+12\right)$
View full question & answer→Question 1012 Marks
Factorise:
$8 a^3-b^3-4 a x+2 b x$
Answer$8 a^3-b^3-4 a x+2 b x$
$=8 a^3-b^3-2 x(2 a-b)$
$=(2 a)^3-(b)^3-2 x(2 a-b) \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=(2 a-b)\left[(2 a)^2+2 a \times b+(b)^2\right]-2 x(2 a-b)$
$=(2 a-b)\left(4 a^2+2 a b+b^2\right)-2 x(2 a-b)$
$=(2 a-b)\left(4 a^2+2 a b+b^2-2 x\right)$
View full question & answer→Question 1022 Marks
Factorise:
$a^2 x^2+\left(a x^2+1\right) x+a$
Answer$a^2 x^2+\left(a x^2+1\right) x+a$
$=a^2 x^2+a x^3+x+a$
$=a x^2(a+x)+1(x+a)$
$=\left(a x^2+1\right)(a+x)$
View full question & answer→Question 1032 Marks
Factorise:
$4 x^2-9 y^2-2 x-3 y$
Answer$4 x^2-9 y^2-2 x-3 y$
$=(2 x)^2-(3 y)^2-(2 x+3 y)\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=(2 x+3 y)(2 x-3 y)-(2 x+3 y)$
$=(2 x+3 y)(2 x-3 y-1)$
View full question & answer→Question 1042 Marks
Factorise:
$6x^2 - 5x - 21$
Answer$6x^2 - 5x - 21$
$= 6x^2 + 9x - 14x - 21$
$= 3x(2x + 3) - 7(2x + 3)$
$= (3x - 7)(2x + 3)$
View full question & answer→Question 1052 Marks
Factorise:
$a - b - a^2 + b^2$
Answer$a - b - a^2 + b^2$
$= (a - b) - (a^2 - b^2)$
$= (a - b) - (a - b)(a + b)$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
$= (a - b)(1 - a - b)$
View full question & answer→Question 1062 Marks
Factorise:
$4a^2 - 9b^2 - 2a - 3b$
Answer$4a^2 - 9b^2 - 2a - 3b$
$= (2a)^2 - (3b)^2 - (2a + 3b)$
$= (2a - 3b)(2a + 3b) - (2a + 3b)$
$= (2a + 3b)(2a - 3b - 1)$
View full question & answer→Question 1072 Marks
Factorise:
$x^2 + y^2 - z^2 - 2xy$
Answer$x^2 + y^2 - z^2 - 2xy$
$= (x^2 + y^2 - 2xy) - z^2$
$= (x - y)^2 - z^2$
$= (x - y - z)(x - y + z)$
View full question & answer→Question 1082 Marks
Factorise:
$x^5 + x^2$
Answer$x^5 + x^2$
$= x^2(x^3 + 1)$
$= x^2(x + 1)[(x)^2 - x \times 1 + (1)^2] Since a^3 + b^3 = (a + b)(a^2 - a \times b + b^2)$
$= x^2(x + 1)(x^2 - x + 1)$
View full question & answer→Question 1092 Marks
Factorise:
$x^2 + 2xy + y^2 - a^2 + 2ab - b^2$
Answer$x^2 + 2xy + y^2 - a^2 + 2ab - b^2$
$= (x^2 + 2xy + y^2) - (a^2 - 2ab + b^2)$
$= (x + y)^2 - (a - b)^2$
$= [(x + y) - (a - b)][(x + y) + (a - b)]$
$= (x + y - a + b)(x + y + a - b)$
View full question & answer→Question 1102 Marks
Factorise:
$a^2+a b(b+1)+b^3$
Answer$a^2+a b(b+1)+b^3$
$=a^2+a b^2+a b+b^3$
$=a^2+a b+a b^2+b^3$
$=a(a+b)+b^2(a+b)$
$=(a+b)\left(a+b^2\right)$
View full question & answer→Question 1112 Marks
Expand:$\Big(\frac{1}{2}\text{a}-\frac{1}{4}\text{b}+2\Big)^2$
Answer$\Big(\frac{1}{2}\text{a}-\frac{1}{4}\text{b}+2\Big)^2=\Big[\Big(\frac{\text{a}}{2}\Big)+\Big(-\frac{\text{b}}{4}\Big)+(2)\Big]^2$$=\Big(\frac{\text{a}}{2}\Big)^2+\Big(-\frac{\text{b}}{4}\Big)^2+(2)^2\\+2\Big(\frac{\text{a}}{2}\Big)\times\Big(\frac{-\text{b}}{4}\Big)(2)+2\Big(\frac{\text{a}}{2}\Big)(2)$
$=\frac{\text{a}^2}{4}+\frac{\text{b}^2}{16}+4-\frac{\text{ab}}{4}-\text{b}+2\text{a}$
View full question & answer→Question 1122 Marks
Factorise:
$3 a^7 b-81 a^4 b^4$
Answer$3 a^7 b-81 a^4 b^4$
$=3 a^4 b\left(a^3-27 b^3\right)$
$=3 a^4 b\left[(a)^3-(3 b)^3\right]$
$=3 a^4 b(a-3 b)\left[(a)^2+a \times 3 b+(3 b)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=3 a^4 b(a-3 b)\left(a^2+3 a b+9 b^2\right)$
View full question & answer→Question 1132 Marks
Factorise:
$9a^2 + 6a + 1 - 36b^2$
Answer$9a^2 + 6a + 1 - 36b^2$
$= (9a^2 + 6a + 1) - 36b^2$
$= [(3a)^2 + 2(3a)(1) + (1)^2] - (6b)^2$
$= (3a + 1)^2 - (6b)^2$
$= (3a + 1 - 6b)(3a + 1 + 6b)$
View full question & answer→Question 1142 Marks
Factorise:
$8-27 b^3-343 c^3-126 b c$
Answer$8-27 b^3-343 c^3-126 b c$
$=(2)^3+(-3 b)^3+(-7 c)^3-3 \times(2) \times(-3 b) \times(-7 c)$
$=\left[2+(-3 b)+(-7 c)\left[(2)^2+(-3 b)^2+(-7 c)^2-(2)(-3 b)-(-3 b)(-7 c)-(2)(-7 c)\right]\right.$
$=(2-3 b-7 c)\left(4+9 b^2+49 c^2+6 b-21 b c+14 c\right)$
View full question & answer→Question 1152 Marks
Find the product.
$(x - 2y + 3)(x^2 + 4y^2 + 2xy + 6y - 3x + 9)$
Answer$(x - 2y + 3)(x^2 + 4y^2 + 2xy + 6y - 3x + 9)= (x - 2y + 3)(x^2 + 4y^2 + 9 + 2xy + 6y - 3x)$
$= [x + (-2y) + 3][x^2 + (-2y)^2 + (3)^2 - x \times (-2y) - (-2y) \times 3 - 3 \times x]$
$= (x)^3 + (-2y)^3 + (3)^3 - 3(x)(-2y)(3)$
$= x^3 - 8y^3 + 27 + 18xy$
View full question & answer→Question 1162 Marks
Factorise:
$108a^2 - 3(b - c)^2$
Answer$108a^2 - 3(b - c)^2$
$= 3[(36a^2 - (b -c)^2]$
$= 3[(6a)^2 - (b - c)^2]$ $\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
$= 3(6a + b - c)(6a - b + c)$
View full question & answer→Question 1172 Marks
Expand:$\Big(\frac{4}{5}\text{a}-2\Big)^3$
Answer$\Big(\frac{4}{5}\text{a}-2\Big)^3$$\Big(\frac{4}{5}\text{a}\Big)^3-(2)^3-3\Big(\frac{4}{5}\text{a}\Big)^2(2)+3\Big(\frac{4}{5}\text{a}\Big)(2)^2$
$=\frac{64}{125}\text{a}^3-8-\frac{96}{25}\text{a}^2+\frac{48}{5}\text{a}$
View full question & answer→Question 1182 Marks
Factorise:
$x^3 - 3x^2+ 3x + 7$
Answer$x^3 - 3x^2+ 3x + 7$
$= x^3 - 3x^2+ 3x - 1 + 8$
$= (x^3 - 3x^2+ 3x - 1) + 8$
$= (x - 1)^3 + 2^3$
$= (x - 1 + 2)[(x - 1)^2 - (x - 1)(2) + 2^2]$
$= (x + 1)(x^2 - 2x + 1 - 2x + 2 + 4)$
$= (x + 1)(x^2 - 4x + 7)$
View full question & answer→Question 1192 Marks
Factorise:
$x^2 + 19x - 150$
Answer$x^2 + 19x - 150$
$= x^2 + 25x - 6x - 150$
$= x(x + 25) - 6(x + 25)$
$= (x + 25)(x - 6)$
View full question & answer→Question 1202 Marks
Factorise:
$(x + 2)^3 - (x - 2)^3$
Answer$(x + 2)^3 - (x - 2)^3$
$= [(x + 2) - (x - 2)][(x + 2)^2 + (x + 2)(x - 2) + (x - 2)^2]$
$= 4(x^2 + 4x + 4 + x^2 - 4 + x^2 - 4x + 4)$
$= 4(3x^2 + 4)$
View full question & answer→Question 1212 Marks
Factorise:
$21x^2 + 5x - 6$
Answer$21x^2 + 5x - 6$
$= 21x^2 + 14x - 9x - 6$
$= 7x(3x + 2) - 3(3x + 2)$
$= (3x + 2)(7x - 3)$
View full question & answer→Question 1222 Marks
Expand:
$(3x + 2)^3$
Answer$(3x + 2)^3$
$= (3x)^3 + 3 \times (3x)^2x^2 + 3 \times 3x \times (2)^2 + (2)^3$
$= 27x^3 + 54x^2 + 36x + 8$
View full question & answer→Question 1232 Marks
Factorise:
$x^2 - 22x + 120$
Answer$x^2 - 22x + 120$
$= x^2 - 10x - 12x + 120$
$= x(x - 10) - 12(x - 10)$
$= (x - 10)(x - 12)$
View full question & answer→Question 1242 Marks
Factorise:
$x(x+y)^3-3 x^2 y(x+y)$
Answer$x(x+y)^3-3 x^2 y(x+y)$
$=x(x+y)\left[(x+y)^2-3 x y\right]$
$=x(x+y)\left(x^2+y^2+2 x y-3 x y\right)$
$=x(x+y)\left(x^2+y^2-x y\right)$
View full question & answer→Question 1252 Marks
Factorise:
$16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z$
AnswerWe have:
$16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z$
$=(4 x)^2+(-2 y)^2+(3 z)^2+2(4 x)(-2 y)+2(-2 y)(3 z)+2(3 z)(4 x)$
$=(4 x-2 y+3 z)^2\left[\text { using } a^2+b^2+c^2+2 a b+2 b c+2 c a=(a+b+c)^2\right]$
Hence, $16 x^2+4 y^2+9 z^2-16 x y-12 y z+24 x z=(4 x-2 y+3 z)^2$
View full question & answer→Question 1262 Marks
Factorise:
$9 x^2-3 x-20$
Answer$9 x^2-3 x-20$
$=9 x^2-15 x+12 x-20$
$=3 x(3 x-5)+4(3 x-5)$
$=(3 x-5)(3 x+4)$
View full question & answer→Question 1272 Marks
Factorise:
$1+2 a b-\left(a^2+b^2\right)$
Answer$1+2 a b-\left(a^2+b^2\right)$
$=1-\left(a^2+b^2-2 a b\right)$
$=(1)^2-(a-b)^2$
$=[1-(a-b)][1+(a-b)]$
$=(1-a+b)(1+a-b)$
View full question & answer→Question 1282 Marks
Factorise:
$216+27 b^3+8 c^3-108 b c$
Answer$216+27 b^3+8 c^3-108 b c$
$=(6)^3+(3 b)^3+(2 c)^3-3 \times 6 \times 3 b \times 2 c$
$=(6+3 b+2 c)\left[6^2+(3 b)^2+(2 c)^2-6 \times 3 b-3 b \times 2 c-2 c \times 6\right]$
$=(6+3 b+2 c)\left(36+9 b^2+4 c^2-18 b-6 b c-12 c\right)$
View full question & answer→Question 1292 Marks
Factorise:
$x^4 y^4-x y$
Answer$x^4 y^4-x y$
$=x y\left(x^3 y^3-1\right)$
$=x y\left[(x y)^3-(1)^3\right]$
$=x y\left\{(x y-1)\left[(x y)^2+(x y)(1)+(1)^2\right]\right\}$
$=x y(x y-1)\left(x^2 y^2+x y+1\right)$
View full question & answer→Question 1302 Marks
Factorise:
$x^2-y^2+6 y-9$
Answer$x^2-y^2+6 y-9$
$=x^2-\left(y^2-6 y+9\right)$
$=x^2-\left(y^2-2 \times y \times 3+3^2\right)$
$=x^2-(y-3)^2\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=[x+(y-3)][x-(y-3)]$
$=(x+y-3)(x-y+3)$
View full question & answer→Question 1312 Marks
Factorise:$\text{x}^2-\sqrt{3}\text{x}-6$
Answer$\text{x}^2-\sqrt{3}\text{x}-6$$=\text{x}^2-2\sqrt{3}\text{x}+\sqrt{3}\text{x}-6$
$=\text{x}(\text{x}-2\sqrt{3})+\sqrt{3}(\text{x}-2\sqrt{3})$
$=(\text{x}-2\sqrt{3})(\text{x}+\sqrt{3})$
View full question & answer→Question 1322 Marks
Factorise:
$(a+2 b)^2+101(a+2 b)+100$
AnswerGiven equation: $(a+2 b)^2+101(a+2 b)+100$
Let $(a+2 b)=x$
Then, we have
$x^2+101 x+100$
$=x^2+100 x+x+100$
$=x(x+100)+1(x+100)$
$=(x+100)(x+1)$
$=(a+2 b+100)(a+2 b+1)$
View full question & answer→Question 1332 Marks
Factorise:
$1+b^3+8 c^3-6 b c$
Answer$1+b^3+8 c^3-6 b c$
$=(1)^3+(b)^3+(2 c)^3-3 \times 1 \times b \times 2 c$
$=(1+b+2 c)\left[1^2+b^2+(2 c)^2-1 \times b-b \times 2 c-1 \times 2 c\right]$
$=(1+b+2 c)\left(1^2+b^2+4 c^2-b-2 b c-2 c\right)$
View full question & answer→Question 1342 Marks
Factorise:
$16 x^4-1$
Answer$16 x^4-1$
$=\left(4 x^2\right)^2-(1)^2$
$=\left(4 x^2-1\right)\left(4 x^2+1\right)$
$=\left[(2 x)^2-(1)^2\right]\left(4 x^2+1\right)$
$=(2 x-1)(2 x+1)\left(4 x^2+1\right)$
View full question & answer→Question 1352 Marks
Factorise:
$3 x^2-14 x+8$
Answer$3 x^2-14 x+8$
$=3 x^2-12 x-2 x+8$
$=3 x(x-4)-2(x-4)$
$=(x-4)(3 x-2)$
View full question & answer→Question 1362 Marks
Factorise:
$2 x^2-7 x-15$
Answer$2 x^2-7 x-15$
$=2 x^2-10 x+3 x-15$
$=2 x(x-5)+3(x-5)$
$=(x-5)(2 x+3)$
View full question & answer→Question 1372 Marks
Factorise:
$16 x^4+54 x$
Answer$16 x^4+54 x$
$=2 x(8 x 3+27)$
$=2 x\left[(2 x)^3+(3)^3\right] \text { Since } a^3+b^3=(a+b)\left(a^2-a \times b+b^2\right)$
$=2 x(2 x+3)\left[(2 x)^2-2 x \times 3+3^2\right]$
$=2 x(2 x+3)\left(4 x^2-6 x+9\right)$
View full question & answer→Question 1382 Marks
Factorise:
$4 a^2-4 b^2+4 a+1$
Answer$4 a^2-4 b^2+4 a+1$
$=\left(4 a^2+4 a+1\right)-4 b^2$
$=\left[(2 a)^2+2 \times 2 a \times 1+(1)^2\right]-(2 b)^2$
$=(2 a+1)^2-(2 b)^2$
$=(2 a+1-2 b)(2 a+1+2 b)$
$=(2 a-2 b+1)(2 a+2 b+1)$
View full question & answer→Question 1392 Marks
Factorise:$\text{x}^2+2\sqrt{3}\text{x}-24$
Answer$\text{x}^2+2\sqrt{3}\text{x}-24$$=\text{x}^2+4\sqrt{3}\text{x}-2\sqrt{3}\text{x}-24$
$=\text{x}(\text{x}+4\sqrt{3})-2\sqrt{3}(\text{x}+4\sqrt{3})$
$=(\text{x}+4\sqrt{3})(\text{x}-2\sqrt{3})$
View full question & answer→Question 1402 Marks
Factorise:
$5 x^2-16 x-21$
Answer$5 x^2-16 x-21$
$=5 x^2+5 x-21 x-21$
$=5 x(x+1)-21(x+1)$
$=(x+1)(5 x-21)$
View full question & answer→Question 1412 Marks
Factorise:Evaluate $\left\{(999)^2-1\right\}$
Answer$\left\{(999)^2-1\right\}$
$=\left\{(999)^2-(1)^2\right\}$
$=\{(999-1)(999+1)\}$
$=998 \times 1000$
$=998000$
View full question & answer→Question 1422 Marks
Factorise:$\text{x}^2+\frac{12}{35}\text{x}+\frac{1}{35}$
Answer$\text{x}^2+\frac{12}{35}\text{x}+\frac{1}{35}$$=\text{x}^2+\frac{5\text{x}}{35}+\frac{\text{x}}{5}+\frac{1}{35}$
$=5\text{x}\Big(\frac{\text{x}}{5}+\frac{1}{35}\Big)+1\Big(\frac{\text{x}}{5}+\frac{1}{35}\Big)$
$=(5\text{x}+1)\Big(\frac{\text{x}}{5}+\frac{1}{35}\Big)$
View full question & answer→Question 1432 Marks
Expand:
$(a-2 b-3 c)^2$
Answer$(a-2 b-3 c)^2=[a+(-2 b)+(-3 c)]^2$
$=(a)^2+(-2 b)^2+(-3 c)^2+2(a)(-2 b)+2(-2 b)(-3 c)+2(a)(-3 c)$
$=a^2+4 b^2+9 c^2-4 a b+12 b c-6 a c$
View full question & answer→Question 1442 Marks
Factorise:
$a^3+3 a^2 b+3 a b^2+b^3-8$
Answer$a^3+3 a^2 b+3 a b^2+b^3-8$
$=(a+b)^3-2^3$
$=[(a+b)-2]\left[(a+b)^2+(a+b) 2+2^2\right]$
$=(a+b-2)\left[(a+b)^2+2(a+b)+4\right]$
View full question & answer→Question 1452 Marks
Factorise:
$x^3-512$
Answer$x^3-512$
$=(x)^3-(8)^3$
$=(x-8)\left[(x)^2+x \times 8+(8)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$
$=(x-8)\left(x^2+8 x+64\right)$
$=x^3+8 x^2+64 x-8 x^2-64 x-512$
$=x^3-512$
View full question & answer→Question 1462 Marks
Factorise:
$x^2-11 x-80$
Answer$x^2-11 x-80$
$=x^2-16 x+5 x-80$
$=x(x-16)+5(x-16)$
$=(x-16)(x+5)$
View full question & answer→Question 1472 Marks
Factorise:
$x^3-x^2+a x+x-a-1$
Answer$x^3-x^2+a x+x-a-1$
$=x^3-x^2+a x-a+x-1$
$=x^2(x-1)+a(x-1)+1(x-1)$
$=(x-1)\left(x^2+a+1\right)$
View full question & answer→Question 1482 Marks
Factorise:
$4(a+b)-6(a+b)^2$
Answer$4(a+b)-6(a+b)^2$
$=(a+b)[4-6(a+b)]$
$=2(a+b)(2-3 a-3 b)$
$=2(a+b)(2-3 a-3 b)$
View full question & answer→Question 1492 Marks
Factorise:$5\sqrt{5}\text{x}^2-20\text{x}+3\sqrt{5}$
Answer$5\sqrt{5}\text{x}^2-20\text{x}+3\sqrt{5}$$=5\sqrt{5}\text{x}^2-15\text{x}-5\text{x}+3\sqrt{5}$
$=5\text{x}\big(\sqrt{5}\text{x}+3\big)+\sqrt{5}\big(\sqrt{5}\text{x}+3\big)$
$=\big(\sqrt{5}\text{x}+3\big)\big(5\text{x}+\sqrt{5}\big)$
View full question & answer→Question 1502 Marks
Factorise:$216\text{x}^3+\frac{1}{125}$
Answer$216\text{x}^3+\frac{1}{125}$We know that:
Since $a^2 + b^3 = (a + b)(a^2 - a \times b + b^2)$
Let us rewrite
$216\text{x}^3+\frac{1}{125}$
$=(6\text{x})^3+\Big(\frac{1}{5}\Big)^3$
$=\Big(6\text{x}+\frac{1}{5}\Big)\bigg[(6\text{x})^2-6\text{x}\times\frac{1}{5}+\Big(\frac{1}{5}\Big)^2\bigg]$
$=\Big(6\text{x}+\frac{1}{5}\Big)\Big(36\text{x}^2-\frac{6\text{x}}{5}+\frac{1}{25}\Big)$
View full question & answer→Question 1512 Marks
Factorise:
$x^2+20 x-69$
Answer$x^2+20 x-69$
$=x^2+23 x-3 x-69$
$=x(x+23)-3(x+23)$
$=(x+23)(x-3)$
View full question & answer→Question 1522 Marks
Factorise:
$15 x^2-x-28$
Answer$15 x^2-x-28$
$=15 x^2+20 x-21 x-28$
$=5 x(3 x+4)-7(3 x+4)$
$=(3 x+4)(5 x-7)$
View full question & answer→Question 1532 Marks
Factorise:
$a^3+8 b^3+64 c^3-24 a b c$
Answer$a^3+8 b^3+64 c^3-24 a b c$
$=a^3+(2 b)^3+(4 c)^3-3 \times a \times 2 b \times 4 c$
$=(a+2 b+4 c)\left[a^2+(2 b)^2+(4 c)^2-a \times 2 b-2 b \times 4 c-4 c \times a\right]$
$=(a+2 b+4 c)\left(a^2+4 b^2+16 c^2-2 a b-8 b c-4 c a\right)$
View full question & answer→Question 1542 Marks
Factorise:
$32 x^4-500 x$
Answer$32 x^4-500 x$
$=4 x\left(8 x^3-125\right)$
$=4 x\left[(2 x)^3-(5)^3\right]$
$=4 x\left[(2 x-5)\left((2 x)^2+2 x \times 5+(5)^2\right] \text { Since } a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)\right.$
$=4 x(2 x-5)\left(4 x^2+10 x+25\right)$
View full question & answer→Question 1552 Marks
Factorise:
$x^2-24 x-180$
Answer$x^2-24 x-180$
$=x^2-30 x+6 x-180$
$=x(x-30)+6(x-30)$
$=(x-30)(x+6)$
View full question & answer→Question 1562 Marks
Evaluate:
$(99)^2$
Answer$(99)^2=(100-1)^2$
$=[(100)+(-1)]^2$
$=(100)^2+2 \times(100) \times(-1)+(-1)^2$
$=10000-200+1$
$=9801$
View full question & answer→Question 1572 Marks
Factorise:$8\text{x}^3-\frac{1}{27\text{y}^3}$
AnswerWe know that: Since $a^3-b^3=(a-b)\left(a^2+a \times b+b^2\right)$ Let us rewrite $8 x^3-\frac{1}{27 y^3}$
$=(2\text{x})^3-\Big(\frac{1}{3\text{y}}\Big)^3$
$=\Big(2\text{x}-\frac{1}{3\text{y}}\Big)\bigg[(2\text{x})^2+2\text{x}\times\frac{1}{3\text{y}}+\Big(\frac{1}{3\text{y}}\Big)^2\bigg]$
$=\Big(2\text{x}-\frac{1}{3\text{y}}\Big)\Big(4\text{x}^2+\frac{2\text{x}}{3\text{y}}+\frac{1}{9\text{y}^2}\Big)$
View full question & answer→Question 1582 Marks
Factorise:
$x^2-26 x+133$
Answer$x^2-26 x+133$
$=x^2-19 x-7 x+133$
$=x(x-19)-7(x-19)$
$=(x-19)(x-7)$
View full question & answer→