## Chapter 9 Algebraic Expressions and Identities Exercise 9.4

Question 1: Multiply the binomials.
i) (2x + 5) and (4x - 3)
ii) (y - 8) and (3y - 4)
iii) (2.5l - 0.5m) and (2.5l + 0.5m)
iv) (a + 3b) and (x + 5)
v) (2pq + 3q²) and (3pq - 2q²)
vi) (3/4 a² + 3b²) and 4(a² - 2/3 b²)

i)
= (2x + 5) × (4x - 3)
= (2x)(4x) - (2x)(3) + (5)(4x) - (5)(3)
= 8x² - 6x + 20x - 15
= 8x² + 14x - 15

ii)
= (y - 8) × (3y - 4)
= (y)(3y) - (y)(4) - (8)(3y) + (8)(4)
= 3y² - 4y - 24y + 32
= 3y² - 28y + 32

iii)
= (2.5l - 0.5m) × (2.5l + 0.5m)
= (2.5l)(2.5l) + (2.5l)(0.5m) - (0.5m)(2.5l) + (0.5m)(0.5m)
= 6.25l² + 1.25lm - 1.25lm + 0.25m²
= 6.25l² + 0.25m²

iv)
= (a + 3b) × (x + 5)
= (a)(x) + (a)(5) + (3b)(x) + (3b)(5)
= ax + 5a + 3bx + 15b

v)
= (2pq + 3q²) × (3pq - 2q²)
= (2pq)(3pq) - (2pq)(2q²) + (3q²)(3pq) - (3q²)(2q²)
= 6p²q² - 4pq³ + 9pq³ - 6q⁴
= 6p²q² + 5pq³ - 6q⁴

vi)
= (3/4 a² + 3b²) × 4(a² - 2/3 b²)
= (3/4 a² + 3b²) × (4a² - 8/3 b²)
= (3/4 a²)(4a²) - (3/4 a²)(8/3 b²) + (3b²)(4a²) + (3b²)(8/3 b²)
= 3a⁴ - 2a²b² + 12b²a² - 8b⁴
= 3a⁴ + 10a²b² - 8b⁴

Question 2: Find the product.
i) (5 - 2x) (3 + x)
ii) (x + 7y) (7x - y)
iii) (a² + b) (a + b²)
iv) (p² - q²) (2p + q)

i)
= (5 - 2x) × (3 + x)
= (5)(3) + (5)(x) - (2x)(3) - (2x)(x)
= 15 + 5x - 6x - 2x²
= 15 - x - 2x²

ii)
= (x + 7y) × (7x - y)
= (x)(7x) - (x)(y) + (7y)(7x) - (7y)(y)
= 7x² - xy + 49xy - 7y²
= 7x² + 48xy - 7y²

iii)
= (a² + b) × (a + b²)
= (a²)(a) + (a²)(b²) + (b)(a) + (b)(b²)
= a³ + a²b² + ab + b³

iv)
= (p² - q²) × (2p + q)
= (p²)(2p) + (p²)(q) - (q²)(2p) - (q²)(q)
= 2p³ + p²q - 2pq² - q³

Question 3: Simplify.
i) (x² - 5) (x + 5) + 25
ii) (a² + 5) (b³ + 3) + 5
iii) (t + s²) (t² - s)
iv) (a + b) (c - d) + (a - b) (c + d) + 2 (ac + bd)
v) (x + y)(2x + y) + (x + 2y)(x - y)
vi) (x + y)(x² - xy + y²)
vii) (1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y
viii) (a + b + c)(a + b - c)

i)
= (x² - 5) (x + 5) + 25
= (x²)(x) + (x²)(5) - (5)(x) - (5)(5) + 25
= x³ + 5x² - 5x - 25 + 25
= x³ + 5x² - 5x

ii)
= (a² + 5) (b³ + 3) + 5
= (a²)(b³) + (a²)(3) + (5)(b³) + (5)(5) + 5
= a²b³ + 3a² + 5b³ + 25 + 5
= a²b³ + 3a² + 5b³ + 20

iii)
= (t + s²) (t² - s)
= (t)(t²) - (t)(s) + (s²)(t²) - (s²)(s)
= t³ - ts + s²t² - s³

iv)
= (a + b) (c - d) + (a - b) (c + d) + 2 (ac + bd)
= (a + b) (c - d) + (a - b) (c + d) + (2ac + 2bd)
= (a)(c) - (a)(d) + (b)(c) - (b)(d) + (a)(c) + (a)(d) - (b)(c) - (b)(d) + 2ac + 2bd
= ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd
= 4ac

v)
= (x + y)(2x + y) + (x + 2y)(x - y)
= (x)(2x) + (x)(y) + (y)(2x) + (y)(y) + (x)(x) - (x)(y) + (2y)(x) - (2y)(y)
= 2x² + xy + 2xy + y² + x² - xy + 2xy - 2y²
= 3x² + 4xy - y²

vi)
= (x + y)(x² - xy + y²)
= (x)(x²) - (x)(xy) + (x)(y²) + (y)(x²) - (y)(xy) + (y)(y²)
= x³ - x²y + xy² + x²y - xy² + y³
= x³ + y³

vii)
= (1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y
= (1.5x)(1.5x) + (1.5x)(4y) + (1.5x)(3) - (4y)(1.5x) - (4y)(4y) - (4y)(3) - 4.5x + 12y
= 2.25x² + 6xy + 4.5x - 6xy - 16y² - 12y - 4.5x + 12y
= 2.25x² - 16y²

viii)
= (a + b + c)(a + b - c)
= (a)(a) + (a)(b) - (a)(c) + (b)(a) + (b)(b) - (b)(c) + (c)(a) + (c)(b) - (c)(c)
= a² + ab - ac + ba + b² - bc + ca + cb - c²
= a² + 2ab + b² - c²