Question 12 Marks
Expand:$(5a - 3b)^3$
Answer$(5a - 3b)^3= (5a)^3 - (3b)^3 - 3(5a)^2(3b) + 3(5a)(3b)^2$
$= 125a^3 - 27b^3 - 225a^2b + 135ab^2$
View full question & answer→Question 22 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 32 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 42 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 52 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 62 Marks
Factorise: $2x^2 - 7x - 15$
Answer$2x^2 - 7x - 15$
$= 2x^2 - 10x + 3x - 15$
$= 2x(x - 5) + 3(x - 5)$
$= (x - 5)(2x + 3)$
View full question & answer→Question 72 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 82 Marks
Factorise: $x^2 - 21x + 90$
Answer$x^2 - 21x + 90$
$= x^2 - 6x - 15x + 90$
$= x(x - 6) - 15(x - 6)$
$= (x - 6)(x - 15)$
View full question & answer→Question 92 Marks
Factorise: $3x^2 - 14x + 8$
Answer$3x^2 - 14x + 8$
$= 3x^2 - 12x - 2x +8$
$= 3x(x - 4) - 2(x - 4)$
$= (x - 4)(3x - 2)$
View full question & answer→Question 102 Marks
Factorise: $2x^2 + 11x - 21$
Answer$2x^2 + 11x - 21$
$= 2x^2 + 14x - 3x - 21$
$= 2x(x + 7) - 3(x + 7)$
$= (x + 7)(2x - 3)$
View full question & answer→Question 112 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 122 Marks
Factorise: $8x^2y^3 - x^5$
Answer$8x^2y^3 - x^5$
$= x^2(8y^3 - x^3)$
$= x^2[(2y)^{3 }- x^3]$
$= x^2[(2y - x)[(2y)^2 + (2y)(x) + x^2]$
$= x^2(2y - x)(4y^2 + 2xy + x^2)$
View full question & answer→Question 132 Marks
Factorise: $6x^2 + 17x + 12$
Answer$6x^2 + 17x + 12$
$= 6x^2 + 9x + 8x + 12$
$= 3x(2x + 3) + 4(2x + 3)$
$= (2x + 3)(3x + 4)$
View full question & answer→Question 142 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 152 Marks
Factorise: $a^2 - b^2 - 4ac + 4c^2$
Answer$a^2 - 4ac + 4c^2 - b^2$
$= a^2 - 4ac + 4c^2 - b^2$
$= a^2 - 2 \times a \times 2c + (2c)^2 - b^2$
$= (a - 2c)^2 - b^2 $$\big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
$= (a - 2c + b)(a - 2c - b)$
View full question & answer→Question 162 Marks
Factorise: $x^2 + 7x - 98$
Answer$x^2 + 7x - 98$
$= x^2 + 14x - 7x - 98$
$= x(x + 14) - 7(x + 14)$
$= (x + 14)(x - 7)$
View full question & answer→Question 172 Marks
Factorise:
$(3a - 2b)^3 + (2b - 5c)^3 + (5c - 3a)^3$
AnswerPut $(3a - 2b) = x, (2b - 5c) = y$ and $(5c - 3a) = z.$
We have:
$x + y + z = 3a - 2b + 2b - 5c + 5c - 3a = 0$
Now,
$(3a - 2b)^3 + (2b - 5c)^3 + (5c - 3a)^3 = x^3 + y^3 + z^3$
$= 3xyz [Here, x + y + z = 0. So, x^3 + y^3 + z^3]$
$= (3a - 2b)(2b - 5c)(5c - 3a)$
View full question & answer→Question 182 Marks
Factorise:$2x^4 - 32$
Answer$2x^4 - 32$
$= 2(x^4 - 16)$
$= 2[(x^2)^2 - (4)^2]$
$= 2[(x^2 - 4)(x^2 + 4)]$
$= 2[(x^2 - 2^2)(x^2 + 4)]$
$= 2[(x - 2)(x + 2)(x^2 + 4)]$
$= 2(x - 2)(x + 2)(x^2 + 4)$
View full question & answer→Question 192 Marks
Factorise:$25x^2^{ }+ 4y^2 + 9z^2 - 20xy - 12yz + 30xz.$
AnswerWe have:
$25x^2 + 4y^2 + 9z^2 - 20xy - 12yz + 30xz$
$= (5x)^2 + (-2y)^2 + (3z)^2 + 2(5x)(-2y) + 2(-2y)(3z) + 2(3z)(5x)$
$= [(5x) + (-2y) + (3z)]^2$
$= (5x - 2y + 3z)^2$
View full question & answer→Question 202 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 212 Marks
Factorise:$64a^3 - 343$
Answer$64a^3 - 343$
$= (4a)^3 - (7)^3$
$= (4a - 7)[(4a)^2 + (4a)(7) + (7)^2]$
$= (4a - 7)(16a^2 + 28a + 49)$
$= 64 + 112 + 196a - 112 - 196a - 343$
$= 64a^3 - 343$
View full question & answer→Question 222 Marks
Factorise:$18x^2 + 3x - 10$
Answer$18x^2 + 3x - 10$
$= 18x^2 - 12x + 15x - 10$
$= 6x(3x - 2) + 5(3x - 2)$
$= (6x + 5)(3x - 2)$
View full question & answer→Question 232 Marks
Factorise:$a^{2 }- b^2 + 2bc - c^2$
Answer$a^{2 }- b^2 + 2bc - c^2$
$= a^2 - (b^2 - 2bc + c^2)$
$= a^2 - (b - c)^2$
$= [a - (b - c)][a + (b - c)]$
$= (a - b + c)(a + b - c)$
View full question & answer→Question 242 Marks
Factorise:$9x^2 + 16y^2 + 4z^2 - 24xy + 16yz - 12xz$
AnswerWe have:
$9x^2 + 16y^2 + 4z^2 - 24xy + 16yz - 12xz$
$= (2x)^2 + (3y)^2 + (-4z)^2 + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x)$
$= [(2x) + (3y) + (-4z)]^2$
$= (2x + 3y - 4z)^2$
View full question & answer→Question 252 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 262 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 272 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 282 Marks
Factorise:$2a^3 + 16b^3 - 5a - 10b$
Answer$2a^3 + 16b^3 - 5a - 10b$
$= 2(a^3 + 8b^3) - 5(a + 2b)$
$= 2[(a)^3 + (2b)^3] - 5(a + 2b)$ Since $a^3 + b^3 = (a + b (a^2 - a \times b + b^2)$
$= 2(a + 2b)[(a)^2 - a \times 2b + (2b)^2] - 5(a + 2b)$
$= (a + 2b)[2(a^2 - 2ab + 4b^2) - 5]$
View full question & answer→Question 292 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 302 Marks
Factorise:$x^6 - 7x^3 - 8$
AnswerGiven equation is $x^6 - 7x^3 - 8.$
Putting $x^3 = y$, we get
$y^2 - 7y - 8$
$= y^2 - 8y + y - 8$
$= y(y - 8) + 1(y - 8)$
$= (y - 8)(y + 1)$
$= (x^3 - 8)(x^3 + 1)$
$= (x^3 - 2^3)(x^3 + 1^3)$
$= (x - 2)(x^2 + 2x + 4)(x + 1)(x^2 - x + 1)$
$= (x - 2)(x + 1)(x^2 + 2x + 4)(x^2 - x + 1)$
View full question & answer→Question 312 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 322 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 332 Marks
Factorise:$2x^2 + 3x - 90$
Answer$2x^2 + 3x - 90$
$= 2x^2 - 12x + 15x - 90$
$= 2x(x - 6) + 15(x - 6)$
$= (x - 6)(2x + 15)$
View full question & answer→Question 342 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 352 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 362 Marks
Factorise:$(a + b)^3 - (a - b)^3$
AnswerWe know that, Since $a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
Therefore,
$(a + b)^3 - (a - b)^3$
$= [a + b - (a - b)][(a + b)^2 + (a + b)(a - b) + (a - b)^2$
$= (a + b - a + b)[a^2 + b^2 + 2ab + a^2 - b^2 + a^2 + b^2 - 2ab]$
$= 2b(3a^2 + b^2)$
View full question & answer→Question 372 Marks
Factorise:$x^2 + 18x + 32$
Answer$x^2 + 18x + 32$
$= x^2 + 16x + 2x + 32$
$= x(x + 16) + 2(x + 16)$
$= (x + 16)(x + 2)$
View full question & answer→Question 382 Marks
Factorise:$8(x + y)^3 - 27(x - y)^3$
Answer$8(x + y)^3 - 27(x - y)^3$
$= [2^{3 }(x + y)^3] - [3^3 (x - y)^3]$
$= [2(x + y) - 3(x - y)]{[2(x + y)]^2 + 2(x + y)3(x - y) + [3(x - y)]^2}$
$= (2x + 2y - 3x + 3y){[4(x^2 + y^2 + 2xy)] + 6(x^2 - y^2) + [9(x^2 + y^2 - 2xy]}$
$= (-x + 5y){4x^2 + 4y^2 + 8xy + 6x^2 - 6y^2 + 9x^2 + 9y^2 - 18xy}$
$= (-x + 5y)(19x^2 + 7y^2 - 10xy)$
View full question & answer→Question 392 Marks
Factorise:$(2a + 1)^3 + (a - 1)^3$
Answer$(2a + 1)^3 + (a - 1)^3$
$= (2a + 1 + a - 1)[(2a + 1)^2 - (2a + 1)(a - 1) + (a - 1)^2]$
$= (3a)[4a^2 + 4a + 1 - 2a^2 + 2a - a + 1 + a^2 - 2a + 1]$
$= 3a(3a^2 + 3a + 3)$
$= 9a(a^2 + a + 1)$
View full question & answer→Question 402 Marks
Factorise:$ab(x^2 + 1) + x(a^2 + b^2)$
Answer$ab(x^2 + 1) + x(a^2 + b^2)$
$= abx^2 + ab + a^2x + b^2x$
$= abx^2 + a^2x + ab + b^2x$
$= ax(bx + a) + b(bx + a)$
$= (bx + a)(ax + b)$
View full question & answer→Question 412 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 422 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 432 Marks
Factorise:$1 - 27a^3$
Answer$1 - 27a^3$
$= (1)^3 - (3a)^3$
$= (1 - 3a)[(1)^2 + 1 \times 3a + (3a)^2]$ Since $a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
$= (1 - 3a)(1 + 3a + 9a^2)$
View full question & answer→Question 442 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 452 Marks
Factorise:$ab(x^2 + y^2) - xy(a^2 + b^2)$
Answer$ab(x^2 + y^2) - xy(a^2 + b^2)$
$= abx^2 + aby^2 - a^2xy - b^2xy$
$= abx^2 - a^2xy + aby^2 - b^2xy$
$= ax(bx - ay) + by(ay - bx)$
$= (bx - ay)(ax - by)$
View full question & answer→Question 462 Marks
Factorise:$x^2 – 32x - 105$
Answer$x^2 – 32x - 105$
$= x^2 – 35x + 3x - 105$
$= x (x - 35) + 3(x - 35)$
$= (x - 35)(x + 3)$
View full question & answer→Question 472 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 482 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 492 Marks
Factorise:$7a^3 + 56b^3$
Answer$7a^3 + 56b^3$
$= 7(a^3 + 8b^3)$
$= 7[(a)^3 + (2b)^3]$
$= 7(a + 2b)[a^2 - a \times 2b + (2b)^2]$ Since $a^3 + b^3 = (a + b)(a^2 - a \times b + b^2)$
$= 7(a + 2b)(a^2 - 2ab + 4b^2)$
View full question & answer→Question 502 Marks
Factorise:$125a^3 + b^3 + 64c^3 - 60abc$
Answer$125a^3 + b^3 + 64c^3 - 60abc$
$= (5a)^3 + (b)^3 + (4c)^3 - 3 \times 5a \times b \times 4c$
$= (5a + b + 4c)[(5a)^2 + (b)^2 + (4c)^2 - 5a \times b - b \times 4c - 5a \times 4c]$
$= (5a + b + 4c)(25a^2 + b^2 + 16c^2 - 5ab - 4bc - 20ac$
View full question & answer→