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:
$6 x^2-5 x-21$
Answer$6 x^2-5 x-21$
$=6 x^2+9 x-14 x-21$
$=3 x(2 x+3)-7(2 x+3)$
$=(3 x-7)(2 x+3)$
View full question & answer→Question 1052 Marks
Factorise:
$a-b-a^2+b^2$
Answer$a-b-a^2+b^2$
$=(a-b)-\left(a^2-b^2\right)$
$=(a-b)-(a-b)(a+b)\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=(a-b)(1-a-b)$
View full question & answer→Question 1062 Marks
Factorise:
$4 a^2-9 b^2-2 a-3 b$
Answer$4 a^2-9 b^2-2 a-3 b$
$=(2 a)^2-(3 b)^2-(2 a+3 b)$
$=(2 a-3 b)(2 a+3 b)-(2 a+3 b)$
$=(2 a+3 b)(2 a-3 b-1)$
View full question & answer→Question 1072 Marks
Factorise:
$x^2+y^2-z^2-2 x y$
Answer$x^2+y^2-z^2-2 x y$
$=\left(x^2+y^2-2 x y\right)-z^2$
$=(x-y)^2-z^2$
$=(x-y-z)(x-y+z)$
View full question & answer→Question 1082 Marks
Factorise:
$x^5+x^2$
Answer$x^5+x^2$
$=x^2\left(x^3+1\right)$
$=x^2(x+1)\left[(x)^2-x \times 1+(1)^2\right] \text { Since } a^3+b^3=(a+b)\left(a^2-a \times b+b^2\right)$
$=x^2(x+1)\left(x^2-x+1\right)$
View full question & answer→Question 1092 Marks
Factorise:
$x^2+2 x y+y^2-a^2+2 a b-b^2$
Answer$x^2+2 x y+y^2-a^2+2 a b-b^2$
$=\left(x^2+2 x y+y^2\right)-\left(a^2-2 a b+b^2\right)$
$=(x+y)^2-(a-b)^2$
$=[(x+y)-(a-b)][(x+y)+(a-b)]$
$=(x+y-a+b)(x+y+a-b)$
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:
$9 a^2+6 a+1-36 b^2$
Answer$9 a^2+6 a+1-36 b^2$
$=\left(9 a^2+6 a+1\right)-36 b^2$
$=\left[(3 a)^2+2(3 a)(1)+(1)^2\right]-(6 b)^2$
$=(3 a+1)^2-(6 b)^2$
$=(3 a+1-6 b)(3 a+1+6 b)$
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.
$(x-2 y+3)\left(x^2+4 y^2+2 x y+6 y-3 x+9\right)$
Answer$(x-2 y+3)\left(x^2+4 y^2+2 x y+6 y-3 x+9\right)$
$=(x-2 y+3)\left(x^2+4 y^2+9+2 x y+6 y-3 x\right)$
$=[x+(-2 y)+3]\left[x^2+(-2 y)^2+(3)^2-x \times(-2 y)-(-2 y) \times 3-3 x x\right]$
$=(x)^3+(-2 y)^3+(3)^3-3(x)(-2 y)(3)$
$=x^3-8 y^3+27+18 x y$
View full question & answer→Question 1162 Marks
Factorise:
$108 a^2-3(b-c)^2$
Answer$108 a^2-3(b-c)^2$
$=3\left[\left(36 a^2-(b-c)^2\right]\right.$
$=3\left[(6 a)^2-(b-c)^2\right]\left[\therefore a^2-b^2=(a-b)(a+b)\right]$
$=3(6 a+b-c)(6 a-b+c)$
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:
$x^3-3 x^2+3 x+7$
Answer$x^3-3 x^2+3 x+7$
$=x^3-3 x^2+3 x-1+8$
$=\left(x^3-3 x^2+3 x-1\right)+8$
$=(x-1)^3+2^3$
$=(x-1+2)\left[(x-1)^2-(x-1)(2)+2^2\right]$
$=(x+1)\left(x^2-2 x+1-2 x+2+4\right)$
$=(x+1)\left(x^2-4 x+7\right)$
View full question & answer→Question 1192 Marks
Factorise:
$x^2+19 x-150$
Answer$x^2+19 x-150$
$=x^2+25 x-6 x-150$
$=x(x+25)-6(x+25)$
$=(x+25)(x-6)$
View full question & answer→Question 1202 Marks
Factorise:
$(x+2)^3-(x-2)^3$
Answer$(x+2)^3-(x-2)^3$
$=[(x+2)-(x-2)]\left[(x+2)^2+(x+2)(x-2)+(x-2)^2\right]$
$=4\left(x^2+4 x+4+x^2-4+x^2-4 x+4\right)$
$=4\left(3 x^2+4\right)$
View full question & answer→Question 1212 Marks
Factorise:
$21 x^2+5 x-6$
Answer$21 x^2+5 x-6$
$=21 x^2+14 x-9 x-6$
$=7 x(3 x+2)-3(3 x+2)$
$=(3 x+2)(7 x-3)$
View full question & answer→Question 1222 Marks
Expand:
$(3 x+2)^3$
Answer$(3 x+2)^3$
$=(3 x)^3+3 \times(3 x)^2 x^2+3 \times 3 x \times(2)^2+(2)^3$
$=27 x^3+54 x^2+36 x+8$
View full question & answer→Question 1232 Marks
Factorise:
$x^2-22 x+120$
Answer$x^2-22 x+120$
$=x^2-10 x-12 x+120$
$=x(x-10)-12(x-10)$
$=(x-10)(x-12)$
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: $\text{x}^2-\sqrt{3}\text{x}-6$
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 1312 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 1322 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 1332 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 1342 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 1352 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 1362 Marks
Factorise:
$16 x^4+54 x$
Answer$16 x^4+54 x$
$=2 x(8 \times 3+27)$
$=2 x\left[(2 x)^3+(3)^3\right] \text { Since } a^3+b^3=(a+b)\left(a^2-a \times b+b^2\right)$
$=2 x(2 x+3)\left[(2 x)^2-2 x \times 3+3^2\right]$
$=2 x(2 x+3)\left(4 x^2-6 x+9\right)$
View full question & answer→Question 1372 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 1382 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 1392 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 1402 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 1412 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 1422 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 1432 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 1442 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 1452 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 1462 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 1472 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 1482 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 1492 Marks
Factorise:
$216\text{x}^3+\frac{1}{125}$
Answer$216\text{x}^3+\frac{1}{125}$
We know that:
Since $a^2+b^3=(a+b)\left(a^2-a \times b+b^2\right)$
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 1502 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→