Worksheet on Algebraic Expressions and Identities

Solved Questions on Algebraic Expression and Identities

1. What is the result if p⁶ − q⁶ is divided by the product of p² + pq+ q²   and p − q?

a) p²− q²
b) p²+ q²
c) p³− q³
d) p³+ q³

Answer: d) p³+ q³

Explanation: Product of p² + pq+ q² and p − q
= (p² + pq+ q²) × (p − q)
= p² (p − q) + pq(p − q)+ q²(p − q)
= p³− p²q+ p²q − pq²+ pq²−q³
= p³− q³

Simplification of p⁶ − q⁶
= (p³)² − (q³)² [Using standard identities a² − b² = (a + b)(a − b)]
= (p³+ q³)(p³ − q³)  

Result = [(p⁶ − q⁶)] ÷ [Product of p² + pq+ q² and p − q]
= [(p³ + q³)(p³ − q³)] ÷ [p³− q³ ]

= p³ + q³

2. What is the length of the adjacent side if an area of a rectangle is (x²−1)(x²−4) and length of one of its sides is (x+1 )(x −2)?

a) x²+x+2
b) x²+x−2
c) x²−x+2
d) x²−x−2

Answer: b) x²+x −2

Explanation:

3. What is the value of st if s + t = 7 and s² + t² = 31?

a) 7
b) 9 
c) 17
d) 19

Answer: b) 9 

Explanation: Using standard identity: (a + b)² = a² + 2ab + b² 

(s + t)² = s² + 2st + t²
⇒ (s + t)² = (s² + t²) + 2st
⇒ (7)² = 31 + 2st
⇒ 49 = 31 + 2st
⇒ 2st = 49 − 31
⇒ 2st = 18
⇒ st = 18/2
⇒ st = 9

4. Simplify: 

a) 
b) 
c) 
d) 

Answer: c) 

Explanation: 

5. What is the value of the given algebraic expression c²d³(c−3d)+7 if c=1/2 and d=2/3? 

a) 6¹¹⁄₂₇
b) 6¹³⁄₂₇
c) 6¹¹⁄₂₇
d) 6¹¹⁄₂₇

Answer: c) 6²⁶⁄₂₇

Explanation: