Question 1012 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 1022 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 1032 Marks
Factorise:$6 - x - x^2$
Answer$6 - x - x^2$
$= 6 + 2x - 3x - x^2$
$= 2(3 + x) - x(3 + x)$
$= (3 + x)(2 - x)$
View full question & answer→Question 1042 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 1052 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 1062 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 1072 Marks
Factorise:$a^2 + ab(b + 1) + b^3$
Answer$a^2 + ab(b + 1) + b^3$
$= a^2 + ab^2 + ab + b^3$
$= a^2 + ab + ab^2 + b^3$
$= a(a + b) + b^2(a + b)$
$= (a + b)(a + b^2)$
View full question & answer→Question 1082 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 1092 Marks
Factorise:$3a^7b - 81a^4b^4$
Answer$3a^7b - 81a^4b^4$
$= 3a^4b(a^3 - 27b^3)$
$= 3a^4b[(a)^3 - (3b)^3]$
$= 3a^4b(a - 3b)[(a)^2 + a \times 3b + (3b)^2] Since a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
$= 3a^4b(a - 3b)(a^2 + 3ab + 9b^2)$
View full question & answer→Question 1102 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 1112 Marks
Factorise:
$8 - 27b^3 - 343c^3 - 126bc$
Answer$8 - 27b^3 - 343c^3 - 126bc$
$= (2)^3 + (-3b)^3 + (-7c)^3 - 3 \times (2) \times (-3b) \times (-7c)$
$= [2 + (-3b) + (-7c)[(2)^2 + (-3b)^2 + (-7c)^2 - (2)(-3b) - (-3b)(-7c) - (2)(-7c)]$
$= (2 - 3b - 7c)(4 + 9b^2 + 49c^2 + 6b - 21bc + 14c)$
View full question & answer→Question 1122 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 1132 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 1142 Marks
Factorise:$x(x + y)^3 - 3x^2y(x + y)$
Answer$x(x + y)^3 - 3x^2y(x + y)$
$= x(x + y)[(x + y)^2 - 3xy]$
$= x(x + y)(x^2 + y^2 + 2xy - 3xy)$
$= x(x + y)(x^2 + y^2 - xy)$
View full question & answer→Question 1152 Marks
Factorise:$16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz$
Answer We have:
$16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz$
$= (4x)^2 + (-2y)^2 + (3z)^2 + 2(4x)(-2y) + 2(-2y)(3z) + 2(3z)(4x)$
$= (4x - 2y + 3z)^2$ [using $a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2]$
Hence, $16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz = (4x - 2y + 3z)^2$
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
Factorise:$1 + 2ab - (a^2 + b^2)$
Answer$1 + 2ab - (a^2 + b^2)$
$= 1 - (a^2 + b^2 - 2ab)$
$= (1)^2 - (a - b)^2$
$= [1 - (a - b)][1 + (a - b)]$
$= (1 - a + b)(1 + a - b)$
View full question & answer→Question 1182 Marks
Factorise: $216 + 27b^3 + 8c^3 - 108bc$
Answer$216 + 27b^3 + 8c^3 - 108bc$
$= (6)^3 + (3b)^3 + (2c)^3 - 3 \times 6 \times 3b \times 2c$
$= (6 + 3b + 2c)[6^2 + (3b)^2 + (2c)^2 - 6 \times 3b - 3b \times 2c - 2c \times 6]$
$= (6 + 3b + 2c)(36 + 9b^2 + 4c^2 - 18b - 6bc - 12c)$
View full question & answer→Question 1192 Marks
Factorise: $x^4y^4 - xy$
Answer$x^4y^4 - xy$
$= xy(x^3y^3 - 1)$
$= xy[(xy)^3 - (1)^3]$
$= xy{(xy - 1)[(xy)^2 + (xy)(1) + (1)^2]}$
$= xy(xy - 1)(x^2y^2 + xy + 1)$
View full question & answer→Question 1202 Marks
Factorise: $x^2 - y^2 + 6y - 9$
Answer$x^2 - y^2 + 6y - 9$
$= x^2 - (y^2 - 6y + 9)$
$= x^2 - (y^2 - 2 \times y \times 3 + 3^2)$
$= x^2 - (y - 3)^2 \big[\therefore\ \text{a}^2-\text{b}^2=(\text{a}-\text{b})(\text{a}+\text{b})\big]$
$= [x + (y - 3)][x - (y - 3)]$
$= (x + y - 3)(x - y + 3)$
View full question & answer→Question 1212 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 1222 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 1232 Marks
Factorise: $(a + 2b)^2 + 101(a + 2b) + 100$
AnswerGiven equation: $(a + 2b)^2 + 101(a + 2b) + 100$
Let $(a + 2b) = x$
Then, we have
$x^2 + 101x + 100$
$= x^2 + 100x + x + 100$
$= x(x + 100) + 1(x + 100)$
$= (x + 100)(x + 1)$
$= (a + 2b + 100)(a + 2b + 1)$
View full question & answer→Question 1242 Marks
Factorise: $1 + b^3 + 8c^3 - 6bc$
Answer$1 + b^3 + 8c^3 - 6bc$
$= (1)^3 + (b)^3 + (2c)^3 - 3 \times 1 \times b \times 2c$
$= (1 + b + 2c)[1^2 + b^2 + (2c)^2 - 1 \times b - b \times 2c - 1 \times 2c]$
$= (1 + b + 2c)(1^2 + b^2 + 4c^2 - b - 2bc - 2c)$
View full question & answer→Question 1252 Marks
Factorise: $16x^4 - 1$
Answer$16x^4 - 1$
$= (4x^2)^2 - (1)^2$
$= (4x^2 - 1)(4x^2 + 1)$
$= [(2x)^2 - (1)^2](4x^2 + 1)$
$= (2x - 1)(2x + 1)(4x^2 + 1)$
View full question & answer→Question 1262 Marks
Factorise:$16x^4 + 54x$
Answer$16x^4 + 54x$
$= 2x(8x 3 + 27)$
$= 2x[(2x)^3 + (3)^3]$ Since $a^3 + b^3 = (a + b)(a^2 - a \times b + b^2)$
$= 2x(2x + 3)[(2x)^2 - 2x \times 3 + 3^2]$
$= 2x(2x+3)(4x^2 -6x +9)$
View full question & answer→Question 1272 Marks
Factorise:$4a^2 - 4b^2 + 4a + 1$
Answer$4a^2 - 4b^2 + 4a + 1$
$= (4a^2 + 4a + 1) - 4b^2$
$= [(2a)^2 + 2 \times 2a \times 1 + (1)^2] - (2b)^2$
$= (2a + 1)^2 - (2b)^2$
$= (2a + 1 - 2b)(2a + 1 + 2b)$
$= (2a - 2b + 1)(2a + 2b + 1)$
View full question & answer→Question 1282 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 1292 Marks
Expand $: (a - 2b - 3c)^2$
Answer$(a - 2b - 3c)^2 $
$= [a + (-2b) + (-3c)]^2 $
$=(a)^2+ (-2b)^2 + (-3c)^2 + 2(a)(-2b) + 2(-2b)(-3c) + 2(a)(-3c)$
$=a^2 + 4b^2 + 9c^2 - 4ab + 12bc - 6ac$
View full question & answer→Question 1302 Marks
Factorise $: a^3 + 3a^2b + 3ab^2 + b^3 - 8$
Answer$a^3 + 3a^2b + 3ab^2 + b^3 - 8$
$= (a + b)^3 - 2^3$
$=[(a + b) - 2][(a + b)^2 + (a + b)2 + 2^2]$
$=(a + b - 2)[(a + b)^2 + 2(a + b) + 4]$
View full question & answer→Question 1312 Marks
Factorise $: x^3 - 512$
Answer$x^3 - 512$
$= (x)^3 - (8)^3$
$= (x - 8)[(x)^2 + x \times 8 + (8)^2]$ Since $a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
$= (x - 8)(x^2 + 8x + 64)$
$= x^3 + 8x^2 + 64x - 8x^2 - 64x - 512$
$= x^3 - 512$
View full question & answer→Question 1322 Marks
Factorise $: x^3 - x^2 + ax + x - a - 1$
Answer$x^3 - x^2 + ax + x - a - 1$
$= x^3 - x^2 + ax - a + x - 1$
$= x^2(x - 1) + a(x - 1) + 1(x - 1)$
$= (x - 1)(x^2 + a + 1)$
View full question & answer→Question 1332 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 1342 Marks
Factorise $: x^2 + 20x - 69$
Answer$x^2 + 20x - 69$
$= x^2 + 23x - 3x - 69$
$= x(x + 23) - 3(x + 23)$
$= (x + 23)(x - 3)$
View full question & answer→Question 1352 Marks
Factorise $: 15x^2 - x - 28$
Answer$15x^2 - x - 28$
$= 15x^2 + 20x - 21x - 28$
$= 5x(3x + 4) - 7(3x + 4)$
$= (3x + 4)(5x - 7)$
View full question & answer→Question 1362 Marks
Factorise $: a^3 + 8b^3 + 64c^3 - 24abc$
Answer$a^3 + 8b^3 + 64c^3 - 24abc$
$= a^3 + (2b)^3 + (4c)^3 - 3 \times a \times 2b \times 4c$
$= (a + 2b + 4c)[a^2 + (2b)^2 + (4c)^2 - a \times 2b - 2b \times 4c - 4c \times a]$
$= (a + 2b + 4c)(a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ca)$
View full question & answer→Question 1372 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 1382 Marks
Factorise : $8\text{x}^3-\frac{1}{27\text{y}^3}$
AnswerWe know that:
Since $a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
Let us rewrite
$8\text{x}^3-\frac{1}{27\text{y}^3}$
$=(2\text{x})^3-\Big(\frac{1}{3\text{y}}\Big)^3$
$=\Big(2\text{x}-\frac{1}{3\text{y}}\Big)\bigg[(2\text{x})^2+2\text{x}\times\frac{1}{3\text{y}}+\Big(\frac{1}{3\text{y}}\Big)^2\bigg]$
$=\Big(2\text{x}-\frac{1}{3\text{y}}\Big)\Big(4\text{x}^2+\frac{2\text{x}}{3\text{y}}+\frac{1}{9\text{y}^2}\Big)$
View full question & answer→Question 1392 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 1402 Marks
Factorise $: 32x^4 - 500x$
Answer$32x^4 - 500x$
$= 4x(8x^3 - 125)$
$= 4x[(2x)^3 - (5)^3]$
$= 4x[(2x - 5)[(2x)^2 + 2x \times 5 + (5)^2]$ Since $a^3 - b^3 = (a - b)(a^2 + a \times b + b^2)$
$= 4x(2x - 5)(4x^2 + 10x + 25)$
View full question & answer→Question 1412 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 1422 Marks
Factorise : $5x^2 - 16x - 21$
Answer$5x^2 - 16x - 21$
$= 5x^2 + 5x - 21x - 21$
$= 5x(x + 1) -21(x + 1)$
$= (x + 1)(5x - 21)$
View full question & answer→Question 1432 Marks
Factorise $: x^2 - 26x + 133$
Answer$x^2 - 26x + 133$
$= x^2 - 19x - 7x + 133$
$= x(x - 19) - 7(x - 19)$
$= (x - 19)(x - 7)$
View full question & answer→Question 1442 Marks
Factorise : Evaluate ${(999)^2 - 1}$
Answer${(999)^2 - 1}$
$= {(999)^2 - (1)^2}$
$= {(999 - 1)(999 + 1)}$
$= 998 \times 1000$
$= 998000$
View full question & answer→Question 1452 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 1462 Marks
Factorise $: 4(a + b) - 6(a + b)^2$
Answer$4(a + b) - 6(a + b)^2$
$= (a + b)[4 - 6(a + b)]$
$= 2(a + b)(2 - 3a - 3b)$
$= 2(a + b)(2 - 3a - 3b)$
View full question & answer→Question 1472 Marks
Factorise $: 9x^2 - 3x - 20$
Answer$9x^2 - 3x - 20$
$= 9x^2 - 15x + 12x - 20$
$= 3x(3x - 5) + 4(3x - 5)$
$= (3x - 5)(3x + 4)$
View full question & answer→Question 1482 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 1492 Marks
Factorise : $x^2 - 24x - 180$
Answer$x^2 - 24x - 180$
$= x^2 - 30x + 6x - 180$
$= x(x - 30) + 6(x - 30)$
$= (x - 30)(x + 6)$
View full question & answer→Question 1502 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→