Question paper, Set 2

$64t^2 - 9 = (8t)^2 - 3^2$

$= (8t + 3)(8t - 3)$

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

Answer: $(8t + 3)(8t - 3)$

$x^2 + 10xz + 25z^2 = x^2 + (5z)^2 + 2(x)(5z)$

$= (x + 5z)^2$

$= (x + 5z)(x + 5z)$

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

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

$25a^2 - 60ac + 36c^2 = (5a)^2 + (6c)^2 - 2(5a)(6c)$

$= (5a - 6c)^2$

$= (5a - 6c)(5a - 6c)$

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

Answer: $(5a - 6c)^2$

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

$= (2x - 1 + 2y)(2x - 1 - 2y)$

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

Answer: $(2x - 1 + 2y)(2x - 1 - 2y)$

$4 \times 40 = 160$: $5 \times 32 = 160$ and $5 + 32 = 37$

$= 4x^2 + 5x + 32x + 40$

$= x(4x + 5) + 8(4x + 5)$

$= (4x + 5)(x + 8)$

Answer: $(4x + 5)(x + 8)$

$6 \times 36 = 216$: $(-8) \times (-27) = 216$ and $-8 - 27 = -35$

$= 6t^2 - 8t - 27t + 36$

$= 2t(3t - 4) - 9(3t - 4)$

$= (3t - 4)(2t - 9)$

Answer: $(3t - 4)(2t - 9)$

$5 \times (-18) = -90$: $3 \times (-30) = -90$ and $3 - 30 = -27$

$= 5b^2 + 3b - 30b - 18$

$= b(5b + 3) - 6(5b + 3)$

$= (5b + 3)(b - 6)$

Answer: $(5b + 3)(b - 6)$

Writing the terms in order: $-2x^2 + 15x - 18$

$(-2) \times (-18) = 36$: $3 \times 12 = 36$ and $3 + 12 = 15$

$= -2x^2 + 3x + 12x - 18$

$= -x(2x - 3) + 6(2x - 3)$

$= (2x - 3)(6 - x)$

Answer: $(2x - 3)(6 - x)$

$12 \times 5 = 60$

$4 \times 15 = 60$ and $4 + 15 = 19$

So the numbers are $4$ and $15$.

Answer: $4$ and $15$

Product $= -18$, sum $= -2\sqrt{6}$: $\sqrt{6} \times (-3\sqrt{6}) = -18$ and $\sqrt{6} + (-3\sqrt{6}) = -2\sqrt{6}$

$= a^2 + \sqrt{6}a - 3\sqrt{6}a - 18$

$= a(a + \sqrt{6}) - 3\sqrt{6}(a + \sqrt{6})$

$= (a + \sqrt{6})(a - 3\sqrt{6})$

Answer: $(a + \sqrt{6})(a - 3\sqrt{6})$

$\displaystyle 4t^2 + \frac{7}{5}t + \frac{1}{10} = \frac{40t^2 + 14t + 1}{10}$

$\displaystyle = \frac{1}{10}(40t^2 + 14t + 1)$

Answer: $\displaystyle \frac{1}{10}(40t^2 + 14t + 1)$

$x^2 + 2y^2 + 32z^2 - 2\sqrt{2}xy + 16yz - 8\sqrt{2}zx = x^2 + (\sqrt{2}y)^2 + (4\sqrt{2}z)^2 - 2(x)(\sqrt{2}y) + 2(\sqrt{2}y)(4\sqrt{2}z) - 2(4\sqrt{2}z)(x)$

$= (-x + \sqrt{2}y + 4\sqrt{2}z)^2$

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

Answer: $(-x + \sqrt{2}y + 4\sqrt{2}z)^2$

$a^3 + 15a(a + 5) + 125 = a^3 + 15a^2 + 75a + 125$

$= a^3 + 5^3 + 3(a)^2(5) + 3(a)(5)^2$

$= (a + 5)^3$

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

Answer: $(a + 5)^3$

$27t^3 + 27t^2 + 9t - 7 = (3t)^3 + 3(3t)^2(1) + 3(3t)(1)^2 + 1^3 - 8$

$= (3t + 1)^3 - 2^3$

$= (3t + 1 - 2)[(3t + 1)^2 + (3t + 1)2 + 2^2]$

$= (3t - 1)[9t^2 + 6t + 1 + 6t + 2 + 4]$

$= (3t - 1)(9t^2 + 12t + 7)$

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

Answer: $(3t - 1)(9t^2 + 12t + 7)$

$64 + 27a^3 = 4^3 + (3a)^3$

$= (4 + 3a)[4^2 - (4)(3a) + (3a)^2]$

$= (3a + 4)(9a^2 - 12a + 16)$

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

Answer: $(3a + 4)(9a^2 - 12a + 16)$

$x^3 - 31x - 30 = x^3 - x - 30x - 30$

$= x(x^2 - 1) - 30(x + 1)$

$= x(x + 1)(x - 1) - 30(x + 1)$

$= (x + 1)[x(x - 1) - 30]$

$= (x + 1)(x^2 - x - 30)$

Now $x^2 - x - 30$: $5 \times (-6) = -30$ and $5 - 6 = -1$

$x^2 - x - 30 = x^2 + 5x - 6x - 30$

$= x(x + 5) - 6(x + 5)$

$= (x + 5)(x - 6)$

So $x^3 - 31x - 30 = (x + 1)(x + 5)(x - 6)$

Answer: $(x + 1)(x + 5)(x - 6)$

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

$= (x^2 + 4z^2)^2 - (2xz)^2$

$= (x^2 + 4z^2 + 2xz)(x^2 + 4z^2 - 2xz)$

$= (x^2 + 2xz + 4z^2)(x^2 - 2xz + 4z^2)$

Answer: $(x^2 + 2xz + 4z^2)(x^2 - 2xz + 4z^2)$

Here $(5a - 4b) + (4b - 3c) + (3c - 5a) = 0$

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

$\therefore (5a - 4b)^3 + (4b - 3c)^3 + (3c - 5a)^3 = 3(5a - 4b)(4b - 3c)(3c - 5a)$

Answer: $3(5a - 4b)(4b - 3c)(3c - 5a)$

$2m^9 - 2mn^8 = 2m(m^8 - n^8)$

$= 2m[(m^4)^2 - (n^4)^2]$

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

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

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

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

Answer: $2m(m^4 + n^4)(m^2 + n^2)(m + n)(m - n)$

$(t + 4)(t^2 + 25) - 10t^2 - 40t = (t + 4)(t^2 + 25) - 10t(t + 4)$

$= (t + 4)(t^2 + 25 - 10t)$

$= (t + 4)(t^2 - 10t + 25)$

$= (t + 4)(t - 5)^2$

Answer: $(t + 4)(t - 5)^2$