Question paper, Set 3

$16m^4 - 625 = (4m^2)^2 - 25^2$

$= (4m^2 + 25)(4m^2 - 25)$

$= (4m^2 + 25)[(2m)^2 - 5^2]$

$= (4m^2 + 25)(2m + 5)(2m - 5)$

Answer: $(4m^2 + 25)(2m + 5)(2m - 5)$

$16x^2 + 40x + 25 = (4x)^2 + 5^2 + 2(4x)(5)$

$= (4x + 5)^2$

$= (4x + 5)(4x + 5)$

using $a^2 + b^2 + 2ab = (a + b)^2$

Answer: $(4x + 5)^2$

$n^2 - 16n + 64 = n^2 + 8^2 - 2(n)(8)$

$= (n - 8)^2$

$= (n - 8)(n - 8)$

using $a^2 + b^2 - 2ab = (a - b)^2$

Answer: $(n - 8)^2$

$9x^2 - 12x + 4 - 25y^2 = (3x - 2)^2 - (5y)^2$

$= (3x - 2 + 5y)(3x - 2 - 5y)$

using $a^2 - 2ab + b^2 = (a - b)^2$ and $a^2 - b^2 = (a + b)(a - b)$

Answer: $(3x - 2 + 5y)(3x - 2 - 5y)$

$2 \times 6 = 12$

$(-1) \times (-12) = 12$ and $-1 - 12 = -13$

So the numbers are $-1$ and $-12$.

Answer: $-1$ and $-12$

$3 \times (-3) = -9$

$(-1) \times 9 = -9$ and $-1 + 9 = 8$

So the numbers are $-1$ and $9$.

Answer: $-1$ and $9$

$29 \times 208 = 6032$

$6032 = 2 \times 2 \times 2 \times 2 \times 13 \times 29 = 26 \times 232$, and $26 + 232 = 258$

$= 29b^2 + 26b + 232b + 208$

$= b(29b + 26) + 8(29b + 26)$

$= (29b + 26)(b + 8)$

Answer: $(29b + 26)(b + 8)$

Writing the terms in order: $-3p^2 + 20p - 32$

$(-3) \times (-32) = 96$: $8 \times 12 = 96$ and $8 + 12 = 20$

$= -3p^2 + 8p + 12p - 32$

$= -p(3p - 8) + 4(3p - 8)$

$= (3p - 8)(4 - p)$

Answer: $(3p - 8)(4 - p)$

Let $2m + n = p$

$(2m + n)^2 - 4(2m + n) - 5 = p^2 - 4p - 5$

$1 \times (-5) = -5$: $1 \times (-5) = -5$ and $1 - 5 = -4$

$= p^2 + p - 5p - 5$

$= p(p + 1) - 5(p + 1)$

$= (p + 1)(p - 5)$

$= [(2m + n) + 1][(2m + n) - 5]$

$= (2m + n + 1)(2m + n - 5)$

Answer: $(2m + n + 1)(2m + n - 5)$

$3 \times 12 = 36$

$(-2) \times (-18) = 36$ and $-2 - 18 = -20$

So the numbers are $-2$ and $-18$.

Answer: $-2$ and $-18$

$3\sqrt{5} \times (-4\sqrt{5}) = 3 \times (-4) \times 5 = -60$: $(-2) \times 30 = -60$ and $-2 + 30 = 28$

$= 3\sqrt{5}t^2 - 2t + 30t - 4\sqrt{5}$

$= t(3\sqrt{5}t - 2) + 2\sqrt{5}(3\sqrt{5}t - 2)$

$= (3\sqrt{5}t - 2)(t + 2\sqrt{5})$

Answer: $(3\sqrt{5}t - 2)(t + 2\sqrt{5})$

$\displaystyle 3t^2 + \frac{13}{30}t - \frac{1}{30} = \frac{90t^2 + 13t - 1}{30}$

$\displaystyle = \frac{1}{30}(90t^2 + 13t - 1)$

$90 \times (-1) = -90$: $(-5) \times 18 = -90$ and $-5 + 18 = 13$

$\displaystyle = \frac{1}{30}[90t^2 - 5t + 18t - 1]$

$\displaystyle = \frac{1}{30}[5t(18t - 1) + 1(18t - 1)]$

$\displaystyle = \frac{1}{30}(18t - 1)(5t + 1)$

$\displaystyle = \left(t + \frac{1}{5}\right)\left(3t - \frac{1}{6}\right)$

Answer: $\displaystyle \frac{1}{30}(18t - 1)(5t + 1)$, i.e. $\displaystyle \left(t + \frac{1}{5}\right)\left(3t - \frac{1}{6}\right)$

Let $a - 2 = x$ and $c + 3 = y$

$3(a - 2)^2 + 17(a - 2)(c + 3) + 10(c + 3)^2 = 3x^2 + 17xy + 10y^2$

$3 \times 10 = 30$: $2 \times 15 = 30$ and $2 + 15 = 17$

$= 3x^2 + 2xy + 15xy + 10y^2$

$= x(3x + 2y) + 5y(3x + 2y)$

$= (3x + 2y)(x + 5y)$

$= [3(a - 2) + 2(c + 3)][(a - 2) + 5(c + 3)]$

$= (3a - 6 + 2c + 6)(a - 2 + 5c + 15)$

$= (3a + 2c)(a + 5c + 13)$

Answer: $(3a + 2c)(a + 5c + 13)$

$9a^2 + 16b^2 + c^2 - 24ab - 8bc + 6ca = (3a)^2 + (4b)^2 + c^2 - 2(3a)(4b) - 2(4b)(c) + 2(c)(3a)$

$= (3a - 4b + c)^2$

using $a^2 + b^2 + c^2 - 2ab - 2bc + 2ca = (a - b + c)^2$

Answer: $(3a - 4b + c)^2$

$p^3 - 6p^2q + 12pq^2 - 8q^3 = p^3 - (2q)^3 - 3(p)^2(2q) + 3(p)(2q)^2$

$= (p - 2q)^3$

using $(a - b)^3 = a^3 - b^3 - 3ab(a - b)$

Answer: $(p - 2q)^3$

$8m^3 - 27n^3 = (2m)^3 - (3n)^3$

$= (2m - 3n)[(2m)^2 + (2m)(3n) + (3n)^2]$

$= (2m - 3n)(4m^2 + 6mn + 9n^2)$

using $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Answer: $(2m - 3n)(4m^2 + 6mn + 9n^2)$

$64x^3 + 1 = (4x)^3 + 1^3$

$= (4x + 1)[(4x)^2 - (4x)(1) + 1^2]$

$= (4x + 1)(16x^2 - 4x + 1)$

using $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Answer: $(4x + 1)(16x^2 - 4x + 1)$

Here $(4x - y) + (y - 5) + (5 - 4x) = 0$

If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$

$\therefore (4x - y)^3 + (y - 5)^3 + (5 - 4x)^3 = 3(4x - y)(y - 5)(5 - 4x)$

Answer: $3(4x - y)(y - 5)(5 - 4x)$

$t^9 - 256t = t(t^8 - 256)$

$= t[(t^4)^2 - 16^2]$

$= t(t^4 + 16)(t^4 - 16)$

$= t(t^4 + 16)[(t^2)^2 - 4^2]$

$= t(t^4 + 16)(t^2 + 4)(t^2 - 4)$

$= t(t^4 + 16)(t^2 + 4)[t^2 - 2^2]$

$= t(t^4 + 16)(t^2 + 4)(t + 2)(t - 2)$

Answer: $t(t^4 + 16)(t^2 + 4)(t + 2)(t - 2)$

$30a^2 + (5a^2 + 6)a + 36 = 30a^2 + 5a^3 + 6a + 36$

$= 5a^3 + 30a^2 + 6a + 36$

$= 5a^2(a + 6) + 6(a + 6)$

$= (a + 6)(5a^2 + 6)$

Answer: $(a + 6)(5a^2 + 6)$