## Chapter 9 Algebraic Expressions and Identities Exercise 9.3

Question 1: Carry out the multiplication of the expressions in each of the following pairs.
i) 4p, q + r
ii) ab, a - b
iii) a + b, 7a²b²
iv) a² - 9, 4a
v) pq + qr + rp, 0

i)
= 4p(q + r)
= 4pq + 4pr

ii)
= ab(a - b)
= a² b - ab²

iii)
= (a + b) × (7a²b²)
= 7a³b² + 7a²b³

iv)
= (a² - 9) × 4a
= 4a² - 36a

v)
= (pq + qr + rp) × 0
= 0

Question 2: Complete the table.

Question 3: Find the product.
i) (a²) × (2a²²) × (4a²⁶)
ii) (2/3 xy) × (-9/10 x²y²)
iii) (-10/3 pq³) × (6/5 p³q)
iv) x × x² × x³ × x⁴

i)
= (a²) × (2a²²) × (4a²⁶)
= 8 × a⁵⁰
= 8a⁵⁰

ii)
= (2/3 xy) × (-9/10 x²y²)
= -18/30 x³y³
= -3/5 x³y³

iii)
= (-10/3 pq³) × (6/5 p³q)
= -60/15 p⁴q⁴
= -4p⁴q⁴

iv)
= x × x² × x³ × x⁴
= x¹⁰

Question 4:
a) Simplify 3x(4x - 5) + 3 and find its values for (i) x = 3 (ii) x = 1/2.
b) Simplify a(a² + a + 1) + 5 and find its value for (i) a = 0, (ii) a = 1 (iii) a = - 1.

a)
= 3x(4x - 5) + 3
= 12x² - 15x + 3
i) x = 3
= 12x² - 15x + 3
= 12 × 3² - 15 × 3 + 3
= 12 × 9 - 45 + 3
= 108 - 45 + 3
= 63 + 3
= 66
ii) x = 1/2
= 12 × (1/2²) - 15 × (1/2) + 3
= 12 × 1/4 - 15/2 + 3
= 3/1 - 15/2 + 3/1
LCM of 1, 2 = 2
= 6/2 + 6/2 - 15/2
= 12/2 - 15/2
= -3/2

b)
= a(a² + a + 1) + 5
= a³ + a² + a + 5
i) x = 0
= a³ + a² + a + 5
= 0³ + 0² + 0 + 5
= 0 + 0 + 0 + 5
= 5
ii) x = 1
= a³ + a² + a + 5
= 1³ + 1² + 1 + 5
= 1 + 1 + 1 + 5
= 8
iii) x = -1
= a³ + a² + a + 5
= (-1³) + (-1²) + (-1) + 5
= -1 + 1 + (-1) + 5
= 4

Question 5:
a) Add: p(p - q), q(q - r) and r(r - p)
b) Add: 2x(z - x - y) and 2y(z - y - x)
c) Subtract: 3l(l - 4m + 5n) from 4l(10n - 3m + 2l)
d) Subtract: 3a(a + b + c ) - 2b(a - b + c) from 4c(-a + b + c)

a)
= p(p - q) + q(q - r) + r(r - p)
= p² - pq + q² - qr + r² - rp
= p² + q² + r² - pq - qr - rp

b)
= 2x(z - x - y) + 2y(z - y - x)
= 2xz - 2x² - 2xy + 2yz - 2y² - 2yx
= 2xz - 2x² - 2xy - 2y² - 2yx + 2yz

c)
= 4l(10n - 3m + 2l) - 3l(l - 4m + 5n)
= 40ln - 12lm + 8l² - 3l² + 12lm - 15ln
= 25ln + 5l²

d)
= 4c(-a + b + c) - [3a(a + b + c ) - 2b(a - b + c)]
= -4ca + 4cb + 4c² - [3a² + 3ab + 3ac - 2ba + 2b² - 2bc]
= -4ca + 4cb + 4c² - 3a² - 3ab - 3ac + 2ba - 2b² + 2bc
= -4ca - 3ac + 4cb + 2bc + 4c² - 3a² - 3ab + 2ba - 2b²
= -7ac + 6bc + 4c² - 3a² - ab - 2b²