Question 13 Marks
Factorise:$x^4 + y^4 - 6x^2y^2$
Answer$x^4 + y^4 - 6x^2y^2$
$= (x^2)^2 + (y)^2 - 2x^2y^2 - 4x^2y^2$
$= [(x^2)^2 + (y)^2 - 2x^2y^2] - (4x^2y^2)$
$= (x^2 - y^2)^2 - (2xy)^2$
$= (x^2- y^2 - 2xy)(x^2 - y^2 + 2xy).$
View full question & answer→Question 23 Marks
Factorise:$4 a^2+\frac{1}{4 a^2}-2-6 a+\frac{3}{2 a}$
Answer$4 a^2+\frac{1}{4 a^2}-2-6 a+\frac{3}{2 a} $
$ =\left(4 a^2+\frac{1}{4 a^2}-2\right)-\left(6 a-\frac{3}{2 a}\right)$
$ =\left(2 a-\frac{1}{2 a}\right)^2-3\left(2 a-\frac{1}{2 a}\right) $
$ =\left(2 a-\frac{1}{2 a}\right)\left(2 a-\frac{1}{2 a}-3\right)$
View full question & answer→Question 33 Marks
Factorise:$y^2+\frac{1}{4 y^2}+1-6 y-\frac{3}{y}$
Answer$y^2+\frac{1}{4 y^2}+1-6 y-\frac{3}{y} $
$=\left(y^2+\frac{1}{4 y^2}+1\right)-\left(6 y+\frac{3}{y}\right) $
$ =\left(y+\frac{1}{2 y}\right)^2-6\left(y+\frac{1}{2 y}\right) $
$=\left(y+\frac{1}{2 y}\right)\left(y+\frac{1}{2 y}-6\right) .$
View full question & answer→Question 43 Marks
Express each of the following as the difference of two squares:$(x^2 + 2x - 3) (x^2 - 2x + 3)$
Answer$(x^2 + 2x - 3) (x^2 - 2x + 3)$
$= [x^2 + (2x - 3)][x^2 - (2x - 3)]$
$= x^2 - (2x - 3)^2$
$= x^4 - (2x - 3)^2.$
View full question & answer→Question 53 Marks
Express each of the following as the difference of two squares:$(x^2 - 2x + 3) (x^2 - 2x - 3)$
Answer$(x^2 - 2x + 3) (x^2 - 2x - 3)$
$= [(x^2 - 2x) + 3][(x^2 - 2x) + 3]$
$= (x^2 - 2x)^2 - (3)^2$
$= (x^2 - 2x)^2 - 9.$
View full question & answer→Question 63 Marks
Express each of the following as the difference of two squares:$(x^2 - 2x + 3)(x^2 + 2x + 3)$
Answer$(x^2 - 2x + 3)(x^2 + 2x + 3)$
$= [x^2 + 3 - 2x][x^2 + 3 + 2x]$
$= [(x^2 + 3) - 2x][(x^2 + 3) + 2x]$
$= (x^2 + 3)^2 - (2x)^2$
$= (x^2 + 3)^2 - 4x^2.$
View full question & answer→Question 73 Marks
Factorise the following:$y^4 + y^2 + 1$
Answer$y^4 + y^2 + 1$
$= y^4 + 2y^2 + 1 - y^2$
$= (y^2 + 1)^2 - y^2$
$= (y^2 + 1 + y)(y^2 + 1 - y).$
View full question & answer→Question 83 Marks
Factorise the following:$(x + y)^3 - x - y$
Answer$(x + y)^3 - x - y$
$= (x + y)(x + y)^2 - (x + y)$
$= (x + y)[(x + y)^2 - 1]$
$= (x + y)[(x + y + 1)(x + y - 1)]$
$= (x + y)(x + y + 1)(x + y - 1).$
View full question & answer→Question 93 Marks
Factorise the following:$4x^2 - 12ax - y^2- z^2 - 2yz + 9a^2$
Answer$4x^2 - 12ax - y^2 - z^2 - 2yz + 9a^2$
$= (4x^2 - 12ax + 9a^2) - (y^2 + z^2 + 2yz)$
$= (2x - 3a)^2 - (y + z)^2$
$= [(2x - 3a) + (y + z)][(2x - 3a) - (y + z)]$
$= (2x - 3a + y + z)(2x - 3a - y - z).$
View full question & answer→Question 103 Marks
Factorise the following:$a^2 + b^2 - c^2 - d^2 + 2ab - 2cd$
Answer$a^2 + b^2 - c^2 - d^2 + 2ab - 2cd$
$= (a^2 + b^2 + 2ab) - (c^2 + d^2 + 2cd)$
$= (a + b)^2 - (c + d)^2$
$= (a + b + c + d)(a + b - c - d).$
View full question & answer→Question 113 Marks
Factorise the following:$x^2+\frac{1}{x^2}-2$
Answer$x^2+\frac{1}{x^2}-2 $
$=x^2+\frac{1}{x^2}-2 x \times x \frac{1}{x} $
$=\left(x-\frac{1}{x}\right)^2 $
$=\left(x-\frac{1}{x}\right)\left(x-\frac{1}{x}\right) .$
View full question & answer→Question 123 Marks
Factorise the following:$b^2 - 2bc + c^2 - a^2$
Answer$b^2 - 2bc + c^2 - a^2$
$= (b^2 - 2bc + c^2) - a^2$
$= (b - c)^2 - (a)^2$
$= (b - c - a)(b - c + a).$
View full question & answer→Question 133 Marks
Factorise the following:$(2a - b)^2 -9(3c - d)^2$
Answer$(2a - b)^2 - 9(3c - d)^2$
$= (2a - b)^2 - [3(3c - d)]^2$
$= [(2a - b) - 3(3c - d)][(2a - b) + 3(3c - d)]$
$= (2a - b - 9c + 3d)(2a - b + 9c - 3d).$
View full question & answer→Question 143 Marks
Factorise the following:$25(x - y)^2 - 49(c - d)^2$
Answer$25(x - y)^2 - 49(c - d)^2$
$= [5(x - y)]^2 - [7(c - )]^2$
$= [5(x - y) - 7(c - d)][5(x - y) + 7(c - d)]$
$= (5x - 5y - 7c + 7d)(5x - 5y + 7c - 7d).$
View full question & answer→Question 153 Marks
Factorise the following by the difference of two squares:$x^2 + y^2 - z^2 - 2xy$
Answer$x^2 + y^2 - z^2 - 2xy$
$= x^2 + y^2 - 2xy - z^2$
$= (x^2 + y^2 - 2xy) - z^2$
$= (x - y)^2 - (z)^2$
$= (x - y - z)(x - y + z).$
View full question & answer→Question 163 Marks
Factorise the following by the difference of two squares:$8xy^2 - 18x^3$
Answer$8xy^2 - 18x^3$
$= 2x(4y^2 -9x^2)$
$= 2x[(2y)^2 - (3x)^2]$
$= 2x[(2y - 3x)(2y + 3x)]$
$= 2x(2y - 3x)(2y + 3x).$
View full question & answer→Question 173 Marks
Factorise the following by the difference of two squares:$441 - 81y^2$
Answer$441 - 81y^2$
$= (21)^2 - (9y)^2$
$= (21 - 9y)(21 + 9y)$
$= 3(7 - 3y)3(7 + 3y)$
$= 9(7 - 3y)(7 + 3y).$
View full question & answer→Question 183 Marks
Factorise the following:$p^4 + 23p^2q^2 + 90q^4$
Answer$P^4 + 23p^2q^2 + 90q^4$
$= p^4 + 18p^2q^2 + 5^2q^2 + 90q^4$
$= p^2(p^2 + 18q^2) + 5q^2(p^2 + 18q^2)$
$= (p^2 + 18q^2)(p^2 + 5q^2).$
View full question & answer→Question 193 Marks
Factorise the following:$(3a - 2b)^2 +3(3a - 2b) - 10$
Answer$(3a - 2b)^2 + 3(3a - 2b) - 10$
$= (3a - 2b)^2 + 5(3a - 2b) - 2(3a - 2b) - 10$
$= (3a - 2b)(3a - 2b + 5) - 2(3a - 2b +5)$
$= (3a - 2b + 5)((3a - 2b - 2).$
View full question & answer→Question 203 Marks
Factorise the following:$5(3x + y)^2 + 6(3x + y) - 8$
Answer$5(3x + y)^2 + 6(3x + y) - 8$
$= 5(3x + y)^2 + 10(3x + y) - 4(3x + y) - 8$
$= 5(3x + y)(3x + y + 2) - 4(3x + y + 2)$
$= (3x + y + 2)[5(3x + y) - 4].$
View full question & answer→Question 213 Marks
Factorise the following:$2 \sqrt{5} x^2-7 x-3 \sqrt{5}$
Answer$2 \sqrt{5} x^2-7 x-3 \sqrt{5} $
$=2 \sqrt{5} x^2-10 x+3 x-3 \sqrt{5} $
$ =2 \sqrt{5} x(x-\sqrt{5})+3(x-\sqrt{5})$
$ =(x-\sqrt{5})(2 \sqrt{5} x+3) .$
View full question & answer→Question 223 Marks
Factorise the following:$6 \sqrt{3} x^2-19 x+5 \sqrt{3}$
Answer$6 \sqrt{3} x^2-19 x+5 \sqrt{3} $
$=6 \sqrt{3} x^2-10 x-9 x+5 \sqrt{3} $
$=2 x(3 \sqrt{3} x-5)-\sqrt{3}(3 \sqrt{3} x-5) $
$=(3 \sqrt{3} x-5)(2 x-\sqrt{3}) .$
View full question & answer→Question 233 Marks
Factorise the following:$y^2 + 3y + 2 + by + 2b$
Answer$y^2 + 3y + 2 + by + 2b$
$= y^2 + y + 2y + 2 + by + 2b$
$= y^2 + y + by + 2y + 2 + 2b$
$= y(y + 1 + b) + 2(y + 1 + b)$
$= (y + 1 + b)(y + 2).$
View full question & answer→Question 243 Marks
Factorise the following:$x^2y^2 + 15xy - 16$
Answer$x^2y^2 + 15xy - 16$
$= x^2y^2 + 16xy - xy - 16$
$= xy(xy + 16) - 1(xy + 16)$
$= (xy + 16)(xy - 1).$
View full question & answer→Question 253 Marks
Factorise the following:$9x^2 - 22xy + 8y^2$
Answer$9x^2 - 22xy + 8y^2$
$= 9x^2 - 18xy - 4xy + 8y^2$
$= 9x(x - 2y) - 4y(x - 2y)$
$= (x - 2y)(9x - 4y).$
View full question & answer→Question 263 Marks
Factorise the following:$5x^2 - 17xy + 6y^2$
Answer$5x^2 - 17xy + 6y^2$
$= 5x^2 - 15xy - 2xy + 6y^2$
$= 5x(x - 3y) - 2y(x - 3y)$
$= (x - 3y)(5x - 2y).$
View full question & answer→Question 273 Marks
Factorise the following by splitting the middle term:$7x^2 + 40x - 12$
Answer$7x^2 + 40x - 12$
$= 7x^2 + 42x - 2x - 12$
$= 7x(x + 6) - 2(x + 6)$
$= (x + 6)(7x - 2).$
View full question & answer→Question 283 Marks
Factorise the following by splitting the middle term:$12 + x - 6x^2$
Answer$12 + x - 6x^2$
$= 12 + 9x -8x - 6x^2$
$= 3(4 + 3x) - 2x(4 + 3x)$
$= (4 + 3x)(3 - 2x).$
View full question & answer→Question 293 Marks
Factorise the following by splitting the middle term:$15a^2 - 14a - 16$
Answer$15a^2 - 14a - 16$
$= 15a^2 - 24a + 10a - 16$
$= 3a(5a - 8) + 2(5a - 8)$
$= (5a - 8)(3a + 2).$
View full question & answer→Question 303 Marks
Factorise the following by splitting the middle term:$3x^2 + 19x - 14$
Answer$3x^2 + 19x - 14$
$= 3x^2 + 21x - 2x - 14$
$= 3x(x + 7) - 2(x + 7)$
$= (x + 7)(3x - 2).$
View full question & answer→Question 313 Marks
Factorise the following by splitting the middle term:$y^2 - 2y - 24$
Answer$y^2 - 2y - 24$
$= y^2 - 6y + 4y - 24$
$= y(y - 6) + 4(y - 6)$
$=(y - 6)(y + 4).$
View full question & answer→Question 323 Marks
Factorise the following by splitting the middle term:$p^2- 12p - 64$
Answer$p^2 - 12p - 64$
$= p^2 - 16p + 4p - 64$
$= p(p - 16) + 4(p - 16)$
$= (p - 16)(p + 4).$
View full question & answer→Question 333 Marks
Factorise the following by splitting the middle term:$x^2 + 5x - 6$
Answer$x^2 + 5x - 6$
$= x^2 + 6x - x - 6$
$= x(x + 6) - 1(x + 6)$
$= (x + 6)(x - 1).$
View full question & answer→Question 343 Marks
Factorise the following by splitting the middle term:$x^2 - 11x + 24$
Answer$x^2 - 11x + 24$
$= x^2 - 8x - 3x + 24$
$= x(x - 8) - 3(x - 8)$
$= (x - 8)(x - 3).$
View full question & answer→Question 353 Marks
Factorise the following by splitting the middle term:$x^2 + 6x + 8$
Answer$x^2 + 6x + 8$
$= x^2 + 4x + 2x + 8$
$= x(x + 4) + 2(x + 4)$
$= (x + 4)(x + 2).$
View full question & answer→Question 363 Marks
Factorise the following by grouping the terms:$2p(a^2 - 2b^2) -14p + (a^2 - 2b^2)^2 - 7(a^2 - 2b^2)$
Answer$2p(a^2 - 2b^2) - 14p + (a^2 - 2b^2)^2 - 7(a^2 - 2b^2)$
$= 2p(a^2 - 2b^2) + (a^2 - 2b^2)^2 - 14p - 7(a^2 - 2b^2)$
$= [2p(a^2 - 2b^2) + (a^2 - 2b^2)^2] - [14p + 7(a^2 - 2b^2)]$
$= (a^2 - 2b^2)(2p + a^2 - 2b^2) - 7(2p + a^2 - 2b^2)$
$= (2p + a^2 - 2b^2)(a^2 - 2b^2 - 7).$
View full question & answer→Question 373 Marks
Factorise the following by grouping the terms:$p^2+\frac{1}{p^2}-2-5 p+\frac{5}{p}$
Answer$p^2+\frac{1}{p^2}-2-5 p+\frac{5}{p} $
$=\left(p^2+\frac{1}{p^2}-2\right)-\left(5 p-\frac{5}{p}\right)$
$=\left((p)^2+\left(\frac{1}{p}\right)^2-2 \times p \times \frac{1}{p}\right)-\left(5 p-\frac{5}{p}\right)$
$=\left(p-\frac{1}{p}\right)^2-5\left(p-\frac{1}{p}\right) $
$ =\left(p-\frac{1}{p}\right)\left(p-\frac{1}{p}-5\right) .$
View full question & answer→Question 383 Marks
Factorise the following by grouping the terms:$p^2x^2 + (px^2 + 1)x + p$
Answer$p^2x^2 + (px^2 + 1)x + p$
$= p^2x^2 + px^3 + x + p$
$= (p^2x^2 + px^3) + (p + x)$
$= px^2(p + x) + 1(p + x)$
$= (p + x)(px^2 + 1).$
View full question & answer→Question 393 Marks
Factorise the following by grouping the terms:$xy(a^2 + 1) + a(x^2 + y^2)$
Answer$xy(a^2 + 1) + a(x^2 + y^2)$
$= a^2xy + xy + ax^2 + ay^2$
$= (a^2xy + ax^2) + (ay^2 + xy)$
$= ax(ay + x) + y(ay + x)$
$= (ay + x)(ax + y).$
View full question & answer→Question 403 Marks
Factorise the following by grouping the terms:$2a + b + 3c - d + (2a + b)^3 + (2a + b)^2(3c - d)$
Answer$2a + b + 3c - d + (2a + b)^3 + (2a + b)^2(3c - d)$
$= (2a + b + 3c - d) + [(2a + b)^3 + (2a + b)^2(3c - d)]$
$= 1(2a + b + 3c - d) + (2a + b)^2(2a + b + 3c - d)$
$= (2a + b + 3c - d)[1 + (2a + b)^2].$
View full question & answer→Question 413 Marks
Factorise the following by grouping the terms:$3ax^2 - 5bx^2 + 9az^2 + 6ay^2 - 10by^2 - 15bz^2$
Answer$3ax^2 - 5bx^2 + 9az^2 + 6ay^2 - 10by^2 - 15bz^2$
$= 3ax^2 + 6ay^2 + 9az^2 - 5bx^2 - 10by^2 - 15z^2$
$= (3ax^2 + 6ay^2 + 9az^2) - (5bx^2 + 10by^2 + 15bz^2)$
$= 3a(x^2 + 2y^2 + 3z^2) - 5b(x^2 + 2y^2 + 3z^2)$
$= (x^2 + 2y^2 + 3z^2)(3a - 5b).$
View full question & answer→Question 423 Marks
Factorise the following by grouping the terms:$9x^3 + 6x^2y^2 - 4y^3 - 6xy$
Answer$9x^3 + 6x^2y^2 - 4y^3 - 6xy$
$= 9x^3 + 6x^2y^2 - 6xy - 4y^3$
$= (9x^3 + 6x^2y^2) - (6xy + 4y^3)$
$= 3x^2(3x + 2y^2) - 2y(3x + 2y^2)$
$= (3x + 2y^2)(3x^2 - 2y).$
View full question & answer→Question 433 Marks
Factorise the following by grouping the terms:$4abx^2 + 49aby^2 + 14xy(a^2 + b^2)$
Answer$4abx^2 + 49aby^2 + 14xy(a^2 + b^2)$
$= 4abx^2 + 49aby^2 + 14a^2xy + 14b^2xy$
$= (4abx^2 + 14a^2xy) + (14b^2xy + 49aby^2)$
$= 2ax(2bx + 7ay) + 7by(2bx + 7ay)$
$= (2bx + 7ay)(2ax + 7by).$
View full question & answer→Question 443 Marks
Factorise the following by grouping the terms:$2m^3 - 5n^2 - 5m^2n + 2mn$
Answer$2m^3 - 5n^2 - 5m^2n + 2mn$
$= 2m^3 + 2mn - 5m^2n - 5n^2$
$= (2m^3 + 2mn) - (5m^2n + 5n^2)$
$= 2m(m^2 + n) - 5n(m^2 + n)$
$= (m^2 + n)(2m - 5n).$
View full question & answer→Question 453 Marks
Factorise the following by grouping the terms:$8(2a + b)^2 - 8a -4b$
Answer$8(2a + b)^2 - 8a - 4b$
$= 8(2a + b)^2 - (8a + 4b)$
$= 8(2a + b)^2 - 4(2a + b)$
$= 4(2a + b)[2(2a + b) - 1]$
$= 4(2a + b)[4a + 2b - 1].$
View full question & answer→Question 463 Marks
Factorise the following by grouping the terms:$9 + 3xy + x^2y + 3x$
Answer$9 + 3xy + x^2y + 3x$
$= 9 + 3xy + 3x + x^2y$
$= (9 + 3xy) + (3x + x^2y)$
$= 3(3 + xy) + y(3 + xy)$
$= (3 + xy)(3 + x).$
View full question & answer→Question 473 Marks
Factorise the following by grouping the terms:$15x^2 + 7y - 3x - 35xy$
Answer$15x^2 + 7y - 3x - 35xy$
$= 15x^2 - 3x - 35xy + 7y$
$= (15x^2 - 3x) - (35xy - 7y)$
$= 3x(5x - 1) - 7y(5x - 1)$
$= (5x - 1)(3x - 7y)$
View full question & answer→Question 483 Marks
Factorise the following by taking out the common factors:$36(x + y)^3 - 54(x + y)^2$
Answer$36(x+y)^3-54(x+y)^2$
Here, the common factor is $18(x+y)^2$.
Dividing throughout by $18(x+y)^2$,
we get
$\frac{36(x+y)^3}{18(x+y)^2}-\frac{54(x+y)^2}{18(x+y)^2} $
$=2(x+y)-3 $
$\therefore 36(x+y)^3-54(x+y)^2 $
$=18(x+y)^2[2(x+y)-3] .$
View full question & answer→Question 493 Marks
Factorise the following by taking out the common factors:$81(p + q)^2 -9p - 9q$
Answer$81(p+q)^2-9 p-9 q $
$=81(p+q)^2-9(p+q)$
Here, the common factor is $9(p+q)$
Dividing throughout by $9(p+q)$,
we get
$\frac{81(p+q)^2}{9(p+q)}-\frac{9(p+q)}{9(p+q)} $
$=9(p+q)-1 $
$\therefore 81(p+q)^2-9 p-9 q $
$=9(p+q)[9(p+q)-1] .$
View full question & answer→Question 503 Marks
Factorise the following by taking out the common factors:$2a(p^2 + q^2) + 4b(p^2 + q^2)$
Answer$2 a\left(p^2+q^2\right)+4 b\left(p^2+q^2\right)$
Here, the common factor is $2\left(p^2+q^2\right)$
Dividing throughtout by $2\left(p^2+q^2\right)$,
we get
$\frac{2 a\left(p^2+q^2\right)}{2\left(p^2+q^2\right)}+\frac{4 b\left(p^2+q^2\right)}{2\left(p^2+q^2\right)} $
$=a+2 b $
$\therefore 2 a\left(p^2+q^2\right)+4 b\left(p^2+q^2\right) $
$=2\left(p^2+q^2\right)(a+2 b) .$
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