Q1. Fill in the blanks to complete the following identities: (i) s2 - 11s + 24 = (_______)(_______) (ii) (_______)(_______) = t2 - 5t + 6 (iii) 10x2 - 11x - 6 = (2x - _______)(_______ + _______) (iv) 6x2 + 7x + 2 = (_______)(_______)
Answer:
(i) s2 - 11s + 24 = (s - 3)(s - 8). Because -3 + (-8) = -11 and (-3)(-8) = 24.
(ii) (t - 2)(t - 3) = t2 - 5t + 6. Because -2 + (-3) = -5 and (-2)(-3) = 6.
(iii) 10x2 - 11x - 6 = (2x - 3)(5x + 2). Because 2x × 5x = 10x2, 2x × 2 + (-3) × 5x = 4x - 15x = -11x, and (-3) × 2 = -6.
(iv) 6x2 + 7x + 2 = (3x + 2)(2x + 1). Because 3x × 2x = 6x2, 3x × 1 + 2 × 2x = 3x + 4x = 7x, and 2 × 1 = 2.
Q2. Select and use the identity that will help you to find the following products without multiplying directly: (i) (41)2 (ii) (27)2 (iii) (23 × 17) (iv) (135)2 (v) (97)2 (vi) (18 × 29) (vii) (34 × 43) (viii) (205)2
Answer:
(i) 412 = (40 + 1)2 = 402 + 2(40)(1) + 12 = 1600 + 80 + 1 = 1681
(ii) 272 = (30 - 3)2 = 302 - 2(30)(3) + 32 = 900 - 180 + 9 = 729
(iii) 23 × 17 = (20 + 3)(20 - 3) = 202 - 32 = 400 - 9 = 391
(iv) 1352 = (100 + 35)2 = 10000 + 2(100)(35) + 352 = 10000 + 7000 + 1225 = 18225
(v) 972 = (100 - 3)2 = 1002 - 2(100)(3) + 32 = 10000 - 600 + 9 = 9409
(vi) 18 × 29 = (18 + 1)(29 - 1) = 19 × 28. Not straightforward. Alternatively, use (20 - 2)(30 - 1) = 600 - 20 - 60 + 2 = 522. Or use 18 × 30 - 18 = 540 - 18 = 522.
(vii) 34 × 43 = (40 - 6)(40 + 3) = 1600 + 120 - 240 - 18 = 1462. Alternatively, 34 × 40 + 34 × 3 = 1360 + 102 = 1462.
(viii) 2052 = (200 + 5)2 = 40000 + 2000 + 25 = 42025
Q3. Factor the following: 9a2 + b2 + 4c2 - 6ab + 12ac - 4bc r2 - r - 42 64u2 + 121v2 + 4w2 - 176uv - 32uw + 44vw
Answer:
9a2 + b2 + 4c2 - 6ab + 12ac - 4bc = (3a)2 + (-b)2 + (2c)2 + 2(3a)(-b) + 2(-b)(2c) + 2(2c)(3a) = (3a - b + 2c)2
r2 - r - 42 = r2 - 7r + 6r - 42 = r(r - 7) + 6(r - 7) = (r - 7)(r + 6)
64u2 + 121v2 + 4w2 - 176uv - 32uw + 44vw = (8u)2 + (-11v)2 + (2w)2 + 2(8u)(-11v) + 2(-11v)(2w) + 2(2w)(8u) = (8u - 11v + 2w)2