1. **Problem 1: Factorizing and finding values in given polynomials** We use special factorization formulas: - Difference of cubes: $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ - Perfect square trinomial: $$ (mx + n)^2 = m^2x^2 + 2mnx + n^2 $$ - Difference of squares: $$a^2 - b^2 = (a - b)(a + b)$$ **(1)** Given $$ax^3 - 27 = (2x - 3)(4x^2 + bx + 9)$$ Expand right side: $$ (2x)(4x^2) + (2x)(bx) + (2x)(9) - 3(4x^2) - 3(bx) - 3(9) = 8x^3 + 2bx^2 + 18x - 12x^2 - 3bx - 27 $$ Group like terms: $$ 8x^3 + (2b - 12)x^2 + (18 - 3b)x - 27 $$ Match coefficients with left side $$ax^3 - 27$$: - Coefficient of $$x^3$$: $$a = 8$$ - Coefficient of $$x^2$$: $$2b - 12 = 0 \\ \Rightarrow 2b = 12 \\ b = 6$$ - Coefficient of $$x$$: $$18 - 3b = 0 \\ 18 - 3(6) = 0$$ (checks out) Calculate $$ab = 8 \times 6 = 48$$ Answer: (d) 48 **(2)** Polynomial $$kx^2 - 12x + 9$$ is a perfect square trinomial. Form of perfect square trinomial: $$ (mx - n)^2 = m^2x^2 - 2mnx + n^2 $$ Match terms: - $$k = m^2$$ - $$-12 = -2mn$$ - $$9 = n^2$$ From $$9 = n^2$$, $$n = \pm 3$$ From $$-12 = -2mn$$, $$12 = 2mn \Rightarrow 6 = mn$$ If $$n = 3$$, then $$6 = 3m \Rightarrow m = 2$$ If $$n = -3$$, then $$6 = -3m \Rightarrow m = -2$$ Calculate $$k = m^2 = (\pm 2)^2 = 4$$ Answer: (b) ±4 **(3)** Given $$ax^2 - b = (3x - 2)(3x + 2)$$ Right side is difference of squares: $$ (3x)^2 - 2^2 = 9x^2 - 4 $$ Match terms: $$ a = 9, \quad b = 4 $$ Calculate $$a + b = 9 + 4 = 13$$ Answer: (b) 13 **(4)** Polynomial $$4x^2 + kx + 1$$ is a perfect square trinomial. Form: $$ (mx + n)^2 = m^2x^2 + 2mnx + n^2 $$ Match terms: - $$4 = m^2 \Rightarrow m = \pm 2$$ - $$1 = n^2 \Rightarrow n = \pm 1$$ - $$k = 2mn$$ Possible values for $$k$$: - If $$m=2, n=1$$, $$k=2 \times 2 \times 1 = 4$$ - If $$m=2, n=-1$$, $$k=2 \times 2 \times (-1) = -4$$ - If $$m=-2, n=1$$, $$k=2 \times (-2) \times 1 = -4$$ - If $$m=-2, n=-1$$, $$k=2 \times (-2) \times (-1) = 4$$ So $$k = \pm 4$$ Answer: (c) ±4 2. **Problem 2: Find $$a$$ to make perfect square trinomials** **(1)** $$x^2 + 14x + a$$ For perfect square: $$ (x + m)^2 = x^2 + 2mx + m^2 $$ Match: $$ 2m = 14 \Rightarrow m = 7 $$ Then: $$ a = m^2 = 7^2 = 49 $$ **(2)** $$x^2 - ax + 64$$ Form: $$ (x - m)^2 = x^2 - 2mx + m^2 $$ Match: $$ -a = -2m \Rightarrow a = 2m $$ $$ 64 = m^2 \Rightarrow m = \pm 8 $$ Calculate $$a$$: - If $$m=8$$, $$a = 2 \times 8 = 16$$ - If $$m=-8$$, $$a = 2 \times (-8) = -16$$ So $$a = \pm 16$$ **Final answers:** Q1: (1) 48 (2) ±4 (3) 13 (4) ±4 Q2: (1) 49 (2) ±16