In Chapter 8, we will solve questions on Algebraic Expressions and Identities. In this chapter of the NCERT book, we will learn skills to handle algebraic expressions, including addition, subtraction, multiplication, and using identities. These algebra concepts will prepare you for the mathematical problem-solving that you will have in senior classes.
Overview of Exercises
Exercise 8.1: Addition and Subtraction of Algebraic Expressions
This exercise will teach us the techniques for adding and subtracting various algebraic expressions. The focus will be to solve questions on handling monomials, binomials, and polynomials, emphasizing the combination and simplification of like terms.
Example: Simplify 3x + 5y – 2x + 4z – 7y + 6z.
Exercise 8.2: Multiplication of Monomials
This exercise will have a multiplication of single-term algebraic expressions, known as monomials. In this, we will understand the role of coefficients and the application of distributive laws in multiplication.
Example: Calculate the product of 5m and 3n.
Exercise 8.3: Multiplication of Binomials
Here, we will explore the multiplication of two-term expressions, or binomials.
Example: Expand and simplify (2y + 5)(3y – 2).
Exercise 8.4: Advanced Multiplication and Algebraic Identities
In this exercise, we will solve questions on advanced multiplication of binomials, trinomials, and polynomials. We will also learn to apply important algebraic identities, such as the difference of squares.
Example: Simplify and express (3x – 4y)(3x + 4y).
- Class 8 Maths Algebraic Expressions and Identities Exercise 8.1
- Class 8 Maths Algebraic Expressions and Identities Exercise 8.2
- Class 8 Maths Algebraic Expressions and Identities Exercise 8.3
- Class 8 Maths Algebraic Expressions and Identities Exercise 8.4
Exercise 8.1 – Class 8 Maths Algebraic Expressions and Identities
1. Add the following:
(i) ab – bc + bc – ca + ca – ab
(ii) a – b + ab + b – c + bc + c – a + ac
(iii) 2p²q² – 3pq + 4 + 5 + 7pq – 3p²q²
(iv) l² + m² + m² + n² + n² + l² + 2lm + 2mn + 2nl
(i) ab – bc + bc – ca + ca – ab
ab – bc + bc – ca + ca – ab
= ab – bc + bc – ca + ca – ab
= 0 (since every term cancels out)
(ii) a – b + ab + b – c + bc + c – a + ac
a – b + ab + b – c + bc + c – a + ac
= (a – a) + (b – b) + (c – c) + ab + bc + ac
= 0 + ab + bc + ac
= ab + bc + ac
(iii) 2p²q² – 3pq + 4 + 5 + 7pq – 3p²q²
2p²q² – 3pq + 4 + 5 + 7pq – 3p²q²
= (2p²q² – 3p²q²) + (-3pq + 7pq) + (4 + 5)
= -p²q² + 4pq + 9
(iv) l² + m² + m² + n² + n² + l² + 2lm + 2mn + 2nl
l² + m² + m² + n² + n² + l² + 2lm + 2mn + 2nl
= (l² + l²) + (m² + m²) + (n² + n²) + 2lm + 2mn + 2nl
= 2l² + 2m² + 2n² + 2lm + 2mn + 2nl
2. Subtract the following:
(a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3
(b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz
(c) Subtract 4p²q – 3pq + 5pq² – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq² + 5p²q
(a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3
(12a – 9ab + 5b – 3) – (4a – 7ab + 3b + 12)
= 12a – 9ab + 5b – 3 – 4a + 7ab – 3b – 12
= (12a – 4a) + (-9ab + 7ab) + (5b – 3b) – (3 + 12)
= 8a – 2ab + 2b – 15
(b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz
(5xy – 2yz – 2zx + 10xyz) – (3xy + 5yz – 7zx)
= 5xy – 2yz – 2zx + 10xyz – 3xy – 5yz + 7zx
= (5xy – 3xy) + (-2yz – 5yz) + (-2zx + 7zx) + 10xyz
= 2xy – 7yz + 5zx + 10xyz
(c) Subtract 4p²q – 3pq + 5pq² – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq² + 5p²q
(18 – 3p – 11q + 5pq – 2pq² + 5p²q) – (4p²q – 3pq + 5pq² – 8p + 7q – 10)
= 18 – 3p – 11q + 5pq – 2pq² + 5p²q – 4p²q + 3pq – 5pq² + 8p – 7q + 10
= (18 + 10) + (-3p + 8p) + (-11q – 7q) + (5pq + 3pq) + (-2pq² – 5pq²) + (5p²q – 4p²q)
= 28 + 5p – 18q + 8pq – 7pq² + p²q
Exercise 8.2 – Class 8 Maths Algebraic Expressions and Identities
1. Find the product of the following pairs of monomials:
(i) 4 × 7p
(ii) –4p × 7p
(iii) –4p × 7pq
(iv) 4p³ × –3p
(v) 4p × 0
(i) 4 × 7p
Product = 4 × 7p
= 28p
(ii) –4p × 7p
Product = –4p × 7p
= –28p²
(iii) –4p × 7pq
Product = –4p × 7pq
= –28p²q
(iv) 4p³ × –3p
Product = 4p³ × –3p
= –12p⁴
(v) 4p × 0
Product = 4p × 0
= 0
2. Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:
(i) p × q
(ii) 10m × 5n
(iii) 20x² × 5y²
(iv) 4x × 3x²
(v) 3mn × 4np
(i) p × q
Area = p × q
= pq
(ii) 10m × 5n
Area = 10m × 5n
= 50mn
(iii) 20x² × 5y²
Area = 20x² × 5y²
= 100x²y²
(iv) 4x × 3x²
Area = 4x × 3x²
= 12x³
(v) 3mn × 4np
Area = 3mn × 4np
= 12mnp²
Exercise 8.3 – Class 8 Maths Algebraic Expressions and Identities
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)
4p × (q + r)
= 4p × q + 4p × r
= 4pq + 4pr
(ii) ab × (a – b)
ab × (a – b)
= ab × a – ab × b
= a²b – ab²
(iii) (a + b) × 7a²b²
(a + b) × 7a²b²
= a × 7a²b² + b × 7a²b²
= 7a³b² + 7ab³
(iv) (a² – 9) × 4a
(a² – 9) × 4a
= a² × 4a – 9 × 4a
= 4a³ – 36a
(v) (pq + qr + rp) × 0
(pq + qr + rp) × 0
= pq × 0 + qr × 0 + rp × 0
= 0 + 0 + 0
= 0
Exercise 8.4 – Class 8 Maths Algebraic Expressions and Identities
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)
(i) (2x + 5) and (4x – 3)
(2x + 5) × (4x – 3)
= 2x × 4x + 2x × (-3) + 5 × 4x + 5 × (-3)
= 8x² – 6x + 20x – 15
= 8x² + 14x – 15
(ii) (y – 8) and (3y – 4)
(y – 8) × (3y – 4)
= y × 3y + y × (-4) – 8 × 3y – 8 × (-4)
= 3y² – 4y – 24y + 32
= 3y² – 28y + 32
(iii) (2.5l – 0.5m) and (2.5l + 0.5m)
(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) and (x + 5)
(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
(v) (2pq + 3q²) and (3pq – 2q²)
(2pq + 3q²) × (3pq – 2q²)
= 2pq × 3pq + 2pq × (-2q²) + 3q² × 3pq – 3q² × 2q²
= 6p²q² – 4pq³ + 9pq³ – 6q⁴
= 6p²q² + 5pq³ – 6q⁴
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 – 2x) × (3 + x)
= 5 × 3 + 5 × x – 2x × 3 – 2x × x
= 15 + 5x – 6x – 2x²
= -2x² – x + 15
(ii) (x + 7y) × (7x – y)
(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 + b) × (a + b²)
= a² × a + a² × b² + b × a + b × b²
= a³ + a²b² + ab + b³
(iv) (p² – q²) × (2p + q)
(p² – q²) × (2p + q)
= p² × 2p + p² × q – q² × 2p – q² × q
= 2p³ + pq² – 2pq² – q³
= 2p³ – pq² – q³
Worksheet for Class 8 Maths Algebraic Expressions and Identities Exercise 8.4
Questions
- Expand and simplify the expression: (3x + 2)² – (2x – 3)².
- Simplify the expression: 2(a + b)(a – b) – a² + b².
- Find the product of (x – 4y) and (x + 4y) and then add the result to x² + 16y².
- If a + b + c = 10 and ab + bc + ca = 21, find the value of a² + b² + c².
- Multiply (3m + 4n)(3m – 4n) and then subtract (5m² – 16n²) from the result.
- Simplify: (x + y + z)² – (x – y + z)² + (x + y – z)².
- Factorize and simplify: 4p²q² – 9r²s².
- Expand the product: (2a – 3b + 4c)(2a + 3b – 4c).
- Simplify the expression: (a + b + c)(a + b – c) + c².
- Find the product of (2x + 3y)(2x – 3y) and add the square of (x + 2y) to the result.
- Expand and simplify: (2p – 3q)³.
- If x + y = 10 and xy = 24, find the value of x² + y².
- Expand and simplify (x – 2y + 3z)(x + 2y – 3z).
- Simplify: (3m² – n²) – 2(m² – 3n²) + 5m².
- Factorize: x⁴ – y⁴.
- Expand and simplify: (x + y)(x – y) + (2x + y)(2x – y).
- If p² + q² = 29 and pq = 10, find the value of (p + q)².
- Factorize and simplify: 25m² – 16n² + 40mn.
- Expand: (3a + 2b – c)(3a – 2b + c).
- Find the product of (5x² – 3y²) and (5x² + 3y²), and then add the square of (2x + 3y) to the result.
Answers
- 20x
- 2a² – 2b² – a² + b² = a² – b²
- 2x²
- 49
- 9m²
- 4xy + 4xz
- (2pq + 3rs)(2pq – 3rs)
- 4a² + 25b² – 16c²
- a² + b² + 2ab
- 4x² + 13y² + 8xy
- 8p³ – 36p²q + 54pq² – 27q³
- 52
- x² – 4y² + 9z²
- 6m² + 5n²
- (x² + y²)(x² – y²)
- 5x² – y²
- 49
- 9m² – 16n² + 40mn = (3m – 4n)(3m + 4n)
- 9a² – c²
- 34x² + 9y²
