1. Factorise the following expressions: **i)** $2b(3a - c) + 12ac - b^2$ - Expand: $6ab - 2bc + 12ac - b^2$ - Group terms: $(6ab + 12ac) - (2bc + b^2)$ - Factor each group: $6a(b + 2c) - b(b + 2c)$ - Factor common binomial: $(6a - b)(b + 2c)$ **ii)** $12(2x - 3y)^2 - 16(3y - 2x)$ - Note $3y - 2x = -(2x - 3y)$ - Rewrite: $12(2x - 3y)^2 + 16(2x - 3y)$ - Factor out $(2x - 3y)$: $(2x - 3y)(12(2x - 3y) + 16)$ - Simplify inside: $(2x - 3y)(24x - 36y + 16)$ **iii)** $y^2 - y(3a - b) - 3ab$ - Expand: $y^2 - 3ay + by - 3ab$ - Group: $(y^2 - 3ay) + (by - 3ab)$ - Factor: $y(y - 3a) + b(y - 3a)$ - Factor common: $(y + b)(y - 3a)$ **iv)** $6ab - b^2 + 12ac - 2bc$ - Group: $(6ab - b^2) + (12ac - 2bc)$ - Factor: $b(6a - b) + 2c(6a - b)$ - Factor common: $(6a - b)(b + 2c)$ **v)** $81x^2 + 90xy + 25y^2$ - Recognize perfect square: $(9x)^2 + 2 imes 9x imes 5y + (5y)^2$ - Factor: $(9x + 5y)^2$ **vi)** $x^4 - y^4$ - Difference of squares: $(x^2)^2 - (y^2)^2 = (x^2 - y^2)(x^2 + y^2)$ - Further factor $x^2 - y^2$: $(x - y)(x + y)(x^2 + y^2)$ **vii)** $a^{12}x^4 - a^4 x^{12}$ - Factor out $a^4 x^4$: $a^4 x^4 (a^8 - x^8)$ - Difference of squares: $a^4 x^4 (a^4 - x^4)(a^4 + x^4)$ - Further factor $a^4 - x^4$: $(a^2 - x^2)(a^2 + x^2)$ - Further factor $a^2 - x^2$: $(a - x)(a + x)$ - Final: $a^4 x^4 (a - x)(a + x)(a^2 + x^2)(a^4 + x^4)$ **viii)** $z^2 + rac{1}{z^2} + 2$ - Recognize perfect square: $(z + rac{1}{z})^2$ **ix)** $9 - 6x + x^2$ - Rearrange: $x^2 - 6x + 9$ - Perfect square: $(x - 3)^2$ **x)** $81 - 4x^2$ - Difference of squares: $(9)^2 - (2x)^2 = (9 - 2x)(9 + 2x)$ **xi)** $m^4 - 256$ - Difference of squares: $(m^2)^2 - 16^2 = (m^2 - 16)(m^2 + 16)$ - Further factor $m^2 - 16$: $(m - 4)(m + 4)(m^2 + 16)$ **xii)** $9x^2 - (2a - 3b)^2$ - Difference of squares: $(3x - (2a - 3b))(3x + (2a - 3b))$ - Simplify: $(3x - 2a + 3b)(3x + 2a - 3b)$ **xiii)** $25 - 4a^2 - 12ab - 9b^2$ - Rearrange: $25 - (4a^2 + 12ab + 9b^2)$ - Recognize perfect square: $25 - (2a + 3b)^2$ - Difference of squares: $(5 - (2a + 3b))(5 + (2a + 3b))$ **xiv)** $x^2 + 11x + 24$ - Find factors of 24 that sum to 11: 8 and 3 - Factor: $(x + 8)(x + 3)$ **xv)** $x^2 + 15x + 56$ - Factors of 56 summing to 15: 7 and 8 - Factor: $(x + 7)(x + 8)$ **xvi)** $x^2 - 2x - 15$ - Factors of -15 summing to -2: -5 and 3 - Factor: $(x - 5)(x + 3)$ **xvii)** $x^2 + x - 12$ - Factors of -12 summing to 1: 4 and -3 - Factor: $(x + 4)(x - 3)$ **xviii)** $4x^2 + 9y^2 + 12xy - 64z^2$ - Group: $(4x^2 + 12xy + 9y^2) - 64z^2$ - Recognize perfect square: $(2x + 3y)^2 - (8z)^2$ - Difference of squares: $(2x + 3y - 8z)(2x + 3y + 8z)$ **xix)** $36p^2 + 6p + rac{1}{4}$ - Multiply entire expression by 4 to check perfect square: $144p^2 + 24p + 1$ - Recognize perfect square: $(12p + 1)^2$ - So original: $(6p + rac{1}{2})^2$ **xx)** $rac{1}{25}x^2 - rac{1}{36}y^2$ - Difference of squares: $(rac{x}{5})^2 - (rac{y}{6})^2 = (rac{x}{5} - rac{y}{6})(rac{x}{5} + rac{y}{6})$ 2. Factorise and divide as directed: i) $\frac{96abc(3a - 12)(5b - 30)}{144(a - 4)(b - 6)}$ - Factor inside parentheses: $3a - 12 = 3(a - 4)$, $5b - 30 = 5(b - 6)$ - Numerator: $96abc \times 3(a - 4) \times 5(b - 6) = 96abc \times 15 (a - 4)(b - 6) = 1440abc (a - 4)(b - 6)$ - Denominator: $144 (a - 4)(b - 6)$ - Cancel common terms: $\frac{1440abc (a - 4)(b - 6)}{144 (a - 4)(b - 6)} = 10abc$ ii) $\frac{44(x^4 - 5x^3 - 24x^2)}{11x(x - 8)}$ - Factor numerator: $x^2(x^2 - 5x - 24)$ - Factor quadratic: $(x - 8)(x + 3)$ - Numerator: $44 x^2 (x - 8)(x + 3)$ - Denominator: $11 x (x - 8)$ - Cancel $11$, $x$, and $(x - 8)$: $4 x (x + 3)$ iii) $\frac{4m^2 - 16mn + 16n^2}{2(m - 2n)}$ - Numerator: $(2m - 4n)^2$ - Simplify numerator: $4(m - 2n)^2$ - Divide: $\frac{4(m - 2n)^2}{2(m - 2n)} = 2(m - 2n)$ iv) $\frac{4yz(z^2 + 6z - 16)}{2y(z + 8)}$ - Factor numerator quadratic: $(z + 8)(z - 2)$ - Numerator: $4 y z (z + 8)(z - 2)$ - Denominator: $2 y (z + 8)$ - Cancel $2 y (z + 8)$: $2 z (z - 2)$ v) $\frac{39 y^3 (50 y^2 - 98)}{26 y^2 (5y + 7)}$ - Factor numerator bracket: $50 y^2 - 98 = 2(25 y^2 - 49) = 2(5y - 7)(5y + 7)$ - Numerator: $39 y^3 \times 2 (5y - 7)(5y + 7) = 78 y^3 (5y - 7)(5y + 7)$ - Denominator: $26 y^2 (5y + 7)$ - Cancel $26 y^2 (5y + 7)$: $3 y (5y - 7)$ Final answers are provided with detailed steps for each factorisation and division.