Complete Solutions and Summary of Algebraic Expressions and Identities – NCERT Class 8 Mathematics Chapter 8

Comprehensive explanations, examples, and exercises on addition, subtraction, and multiplication of algebraic expressions, monomials, binomials, trinomials, polynomials, and identities from NCERT Class 8 Mathematics Chapter 8.

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Categories: NCERT, Class 8, Mathematics, Algebra, Polynomials, Expressions, Chapter 8

Tags: Algebraic Expressions, Identities, Polynomials, Monomials, Binomials, Trinomials, NCERT Class 8, Mathematics, Chapter 8, Addition, Subtraction, Multiplication, Exercises, Solutions

Algebraic Expressions and Identities - Complete Study Guide

Chapter Overview

Monomial One Term Expression

Binomial Two Terms

Trinomial Three Terms

Polynomial Multiple Terms

What You'll Learn

Addition/Subtraction

Combining like terms in expressions, e.g., \(7x^2 - 4x + 5 + 9x - 10 = 7x^2 + 5x - 5\).

Monomial Multiplication

Multiplying monomials like \(5x \times 4x^2 = 20x^3\).

Monomial by Polynomial

Using distributive law, e.g., \(3x \times (5y + 2) = 15xy + 6x\).

Polynomial by Polynomial

Term-by-term multiplication, e.g., \((2a + 3b) \times (3a + 4b) = 6a^2 + 17ab + 12b^2\).

Historical Context

This chapter builds on basic algebra, introducing operations on expressions. It covers real-world applications like area (\(l \times b\)) and volume (\(l \times b \times h\)), emphasizing distributive law for multiplication.

Key Highlights

Focus on like terms, exponents, and combining results. Examples include patterns of dots for multiplication visualization and practical scenarios like banana pricing: \((p - 2) \times (z - 4)\).

Comprehensive Chapter Summary

1. Addition and Subtraction of Algebraic Expressions

Algebraic expressions are combinations like \(x + 3\), \(2y - 5\). Addition aligns like terms: \(7x^2 - 4x + 5 + 9x - 10 = 7x^2 + 5x - 5\). Subtraction uses additive inverses: \(7x^2 - 4xy + 8y^2 + 5x - 3y - (5x^2 - 4y^2 + 6y - 3) = 2x^2 - 4xy + 12y^2 + 5x - 9y + 3\). More examples include multi-variable terms like \(7xy + 5yz - 3zx + 4yz + 9zx - 4y - 2xy - 3zx + 5x = 5xy + 9yz + 3zx + 5x - 4y\).

2. Multiplication of Algebraic Expressions: Introduction

Patterns and Real-World Applications

Multiplication visualized with dot patterns: \(m \times n\), \((m + 2) \times (n + 3)\). Area: \((l + 5) \times (b - 3)\). Volume: length \(\times\) breadth \(\times\) height. Pricing: \((p - 2) \times (z - 4)\).

Key Formulas

Distributive law: \(a \times (b + c) = ab + ac\). Extended to polynomials.

Additional Content

More formulas: Commutative: \(a \times b = b \times a\). Associative: \((a \times b) \times c = a \times (b \times c)\).

3. Multiplying a Monomial by a Monomial

Two Monomials

\(5x \times 3y = 15xy\), \(5x \times (-3y) = -15xy\), \(5x \times 4x^2 = 20x^3\), \(5x \times (-4xyz) = -20x^2yz\).

Three or More

\(2x \times 5y \times 7z = 70xyz\), \(4xy \times 5x^2y^2 \times 6x^3y^3 = 120x^6y^6\).

Additional Formulas

Exponent rules: \(x^m \times x^n = x^{m+n}\), coefficients multiply directly.

4. Multiplying a Monomial by a Polynomial

By Binomial/Trinomial

\(3x \times (5y + 2) = 15xy + 6x\), \(3p \times (4p^2 + 5p + 7) = 12p^3 + 15p^2 + 21p\).

5. Multiplying a Polynomial by a Polynomial

Binomial by Binomial/Trinomial

\((2a + 3b) \times (3a + 4b) = 6a^2 + 17ab + 12b^2\), \((a + 7) \times (a^2 + 3a + 5) = a^3 + 10a^2 + 26a + 35\).

Additional Content

More examples: \((x - 4) \times (2x + 3) = 2x^2 - 5x - 12\). Formulas for expansion.

6. Key Definitions and Summary

Monomial (one term), binomial (two), trinomial (three), polynomial (multiple). Like terms combine. Multiplication uses distributive law term-by-term.

Questions and Answers from Chapter

Short Questions

Q1. Add: \(ab - bc, bc - ca, ca - ab\).

Answer: \(0\).

Q2. Add: \(a - b + ab, b - c + bc, c - a + ac\).

Answer: \(ab + bc + ac\).

Q3. Subtract: \(4a - 7ab + 3b + 12\) from \(12a - 9ab + 5b - 3\).

Answer: \(8a - 2ab + 2b - 15\).

Q4. Find the product: \(4, 7p\).

Answer: \(28p\).

Q5. Find the product: \(-4p, 7p\).

Answer: \(-28p^2\).

Q6. Find the area for lengths and breadths: \(p, q\).

Answer: \(pq\).

Q7. Obtain the volume: \(5a, 3a^2, 7a^4\).

Answer: \(105a^7\).

Q8. Carry out multiplication: \(4p, q + r\).

Answer: \(4pq + 4pr\).

Q9. Find the product: \((a^2) \times (2a^{22}) \times (4a^{26})\).

Answer: \(8a^{49}\).

Q10. Add: \(p(p - q), q(q - r), r(r - p)\).

Answer: \(p^2 - pq + q^2 - qr + r^2 - rp\).

Q11. Multiply the binomials: \((2x + 5)\) and \((4x - 3)\).

Answer: \(8x^2 - 6x + 20x - 15 = 8x^2 + 14x - 15\).

Q12. Simplify: \((x^2 - 5)(x + 5) + 25\).

Answer: \(x^3 + 5x^2 - 5x - 25 + 25 = x^3 + 5x^2 - 5x\).

Q13. Find the product: \((5 - 2x)(3 + x)\).

Answer: \(15 + 5x - 6x - 2x^2 = -2x^2 - x + 15\).

Q14. Simplify: \((t + s^2)(t^2 - s)\).

Answer: \(t^3 - ts + s^2 t^2 - s^3\).

Q15. Simplify: \((a + b + c)(a + b - c)\).

Answer: \(a^2 + 2ab + b^2 - c^2\).

Medium Questions

Q1. Add: \(2p^2 q^2 - 3pq + 4, 5 + 7pq - 3p^2 q^2\).

Answer: Align terms: \(2p^2 q^2 - 3pq + 4 + 5 + 7pq - 3p^2 q^2 = -p^2 q^2 + 4pq + 9\). (3 marks)

Q2. Subtract: \(3xy + 5yz - 7zx\) from \(5xy - 2yz - 2zx + 10xyz\).

Answer: \(5xy - 2yz - 2zx + 10xyz - (3xy + 5yz - 7zx) = 2xy - 7yz + 5zx + 10xyz\). (3 marks)

Q3. Find the areas for: \(10m, 5n\).

Answer: \(10m \times 5n = 50mn\). Similar for others like \(20x^2 \times 5y^2 = 100x^2 y^2\). (3 marks)

Q4. Obtain the product: \(xy, yz, zx\).

Answer: \(xy \times yz \times zx = x^2 y^2 z^2\). (3 marks)

Q5. Carry out multiplication: \(ab, a - b\).

Answer: \(ab(a - b) = a^2 b - ab^2\). (3 marks)

Q6. Simplify \(3x(4x - 5) + 3\) and evaluate for \(x = 3\).

Answer: \(12x^2 - 15x + 3\); for \(x=3\): \(12(9) - 15(3) + 3 = 108 - 45 + 3 = 66\). (3 marks)

Q7. Add: \(2x(z - x - y)\) and \(2y(z - y - x)\).

Answer: \(2xz - 2x^2 - 2xy + 2yz - 2y^2 - 2xy = 2xz + 2yz - 2x^2 - 2y^2 - 4xy\). (3 marks)

Q8. Multiply the binomials: \((y - 8)\) and \((3y - 4)\).

Answer: \(3y^2 - 4y - 24y + 32 = 3y^2 - 28y + 32\). (3 marks)

Q9. Simplify: \((a^2 + 5)(b^3 + 3) + 5\).

Answer: \(a^2 b^3 + 3a^2 + 5b^3 + 15 + 5 = a^2 b^3 + 3a^2 + 5b^3 + 20\). (3 marks)

Q10. Simplify: \((a + b)(c - d) + (a - b)(c + d) + 2(ac + bd)\).

Answer: \(ac - ad + bc - bd + ac + ad - bc - bd + 2ac + 2bd = 4ac + 4bd - 2bd\). Wait, correct: \(2ac + 2bc + 2ac - 2bc + 2ac + 2bd = 4ac + 2bd\). (3 marks)

Q11. Find the product: \((x + 7y)(7x - y)\).

Answer: \(7x^2 - xy + 49xy - 7y^2 = 7x^2 + 48xy - 7y^2\). (3 marks)

Q12. Simplify: \((x + y)(x^2 - xy + y^2)\).

Answer: \(x^3 - x^2 y + x y^2 + x^2 y - x y^2 + y^3 = x^3 + y^3\). (3 marks)

Q13. Simplify: \((1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y\).

Answer: Expand: \(2.25x^2 + 6x - 6x - 16y^2 - 12y + 4.5x - 4.5x + 12y = 2.25x^2 - 16y^2\). (3 marks)

Q14. Carry out multiplication: \(a + b, 7a^2 b^2\).

Answer: \(7a^3 b^2 + 7a^2 b^3\). (3 marks)

Q15. Subtract: \(3a(a + b + c) - 2b(a - b + c)\) from \(4c(-a + b + c)\).

Answer: First expand, then subtract: Detailed steps yield \(-7ac - 3bc + c^2 + 2ab + 2b^2\). (3 marks)

Long Questions

Q1. Add: \(l^2 + m^2, m^2 + n^2, n^2 + l^2, 2lm + 2mn + 2nl\).

Answer: Combine like terms: \(l^2 + m^2 + m^2 + n^2 + n^2 + l^2 + 2lm + 2mn + 2nl = 2l^2 + 2m^2 + 2n^2 + 2lm + 2mn + 2nl\). This is \((l + m + n)^2\), but as per addition, it's the expanded form. Explain step-by-step: First group \(l^2\) terms: \(l^2 + l^2 = 2l^2\), similarly for others, and unlike terms remain.

Q2. Subtract: \(4p^2 q - 3pq + 5pq^2 - 8p + 7q - 10\) from \(18 - 3p - 11q + 5pq - 2pq^2 + 5p^2 q\).

Answer: Align and subtract term by term: \(18 - 3p - 11q + 5pq - 2pq^2 + 5p^2 q - (4p^2 q - 3pq + 5pq^2 - 8p + 7q - 10) = 18 - 3p - 11q + 5pq - 2pq^2 + 5p^2 q - 4p^2 q + 3pq - 5pq^2 + 8p - 7q + 10 = p^2 q + 8pq - 7pq^2 + 5p - 18q + 28\). Detailed explanation of additive inverses and like terms combination.

Q3. Complete the table of products for monomials like \(2x, -5y, 3x^2, -4xy, 7x^2 y, -9x^2 y^2\).

Answer: For example, \(2x \times 2x = 4x^2\), \(2x \times -5y = -10xy\), and so on for all pairs. Explain multiplication rules: coefficients multiply, variables add exponents. Full table construction step-by-step.

Q4. Obtain the volume of rectangular boxes: \(2p, 4q, 8r\).

Answer: \(2p \times 4q \times 8r = 64pqr\). Similarly for others like \(xy \times 2x^2 y \times 2xy^2 = 4x^4 y^4\). Discuss associative property and exponent addition in detail.

Q5. Simplify \(a(a^2 + a + 1) + 5\) and find values for \(a=0,1,-1\).

Answer: \(a^3 + a^2 + a + 5\); for \(a=0\): 5; \(a=1\): 8; \(a=-1\): 4. Step-by-step expansion and substitution explained.

Q6. Multiply the binomials: \((2.5l - 0.5m)\) and \((2.5l + 0.5m)\).

Answer: Difference of squares: \((2.5l)^2 - (0.5m)^2 = 6.25l^2 - 0.25m^2\). Detailed term-by-term multiplication.

Q7. Find the product: \((a + 3b)\) and \((x + 5)\).

Answer: \(a x + 5a + 3b x + 15b\). Explain distributive application over each term.

Q8. Simplify: \((x + y)(2x + y) + (x + 2y)(x - y)\).

Answer: Expand each: \(2x^2 + x y + 2x y + y^2 + x^2 - x y + 2x y - 2y^2 = 3x^2 + 4x y - y^2\). Combine like terms detailed.

Q9. Carry out multiplication: \(pq + qr + rp, 0\).

Answer: \(0\). But explain zero multiplication property in polynomials.

Q10. Complete the table for products like \(x + y - 5, 5xy\).

Answer: \(5x^2 y + 5x y^2 - 25x y\). Step-by-step for each entry in table.

Q11. Find the product: \(x \times x^2 \times x^3 \times x^4\).

Answer: \(x^{10}\). Explain exponent addition rule repeatedly.

Q12. Subtract: \(3l(l - 4m + 5n)\) from \(4l(10n - 3m + 2l)\).

Answer: Expand: \(8l^2 - 12l m + 40l n - (3l^2 - 12l m + 15l n) = 5l^2 + 25l n\). Detailed steps.

Q13. Multiply: \((2pq + 3q^2)\) and \((3pq - 2q^2)\).

Answer: \(6p^2 q^2 - 4p q^3 + 9p q^3 - 6q^4 = 6p^2 q^2 + 5p q^3 - 6q^4\). Combine terms.

Q14. Obtain the product: \(a, -a^2, a^3\).

Answer: \(a \times (-a^2) \times a^3 = -a^6\). Exponent and sign rules explained.

Q15. Simplify: \((a + b + c)abc\).

Answer: \(a^2 bc + ab^2 c + abc^2\). Distributive over trinomial.

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Complete Solutions and Summary of Algebraic Expressions and Identities – NCERT Class 8 Mathematics Chapter 8

Comprehensive explanations, examples, and exercises on addition, subtraction, and multiplication of algebraic expressions, monomials, binomials, trinomials, polynomials, and identities from NCERT Class 8 Mathematics Chapter 8.

Algebraic Expressions and Identities

Chapter Overview

What You'll Learn

Addition/Subtraction

Monomial Multiplication

Monomial by Polynomial

Polynomial by Polynomial

Historical Context

Key Highlights

Comprehensive Chapter Summary

1. Addition and Subtraction of Algebraic Expressions

2. Multiplication of Algebraic Expressions: Introduction

Patterns and Real-World Applications

Key Formulas

Additional Content

3. Multiplying a Monomial by a Monomial

Two Monomials

Three or More

Additional Formulas

4. Multiplying a Monomial by a Polynomial

By Binomial/Trinomial

5. Multiplying a Polynomial by a Polynomial

Binomial by Binomial/Trinomial

Additional Content

6. Key Definitions and Summary

Key Concepts and Definitions

Algebraic Expression

Like Terms

Monomial

Binomial

Trinomial

Polynomial

Important Facts and Figures

Questions and Answers from Chapter

Short Questions

Q1. Add: \(ab - bc, bc - ca, ca - ab\).

Q2. Add: \(a - b + ab, b - c + bc, c - a + ac\).

Q3. Subtract: \(4a - 7ab + 3b + 12\) from \(12a - 9ab + 5b - 3\).

Q4. Find the product: \(4, 7p\).

Q5. Find the product: \(-4p, 7p\).

Q6. Find the area for lengths and breadths: \(p, q\).

Q7. Obtain the volume: \(5a, 3a^2, 7a^4\).

Q8. Carry out multiplication: \(4p, q + r\).

Q9. Find the product: \((a^2) \times (2a^{22}) \times (4a^{26})\).

Q10. Add: \(p(p - q), q(q - r), r(r - p)\).

Q11. Multiply the binomials: \((2x + 5)\) and \((4x - 3)\).

Q12. Simplify: \((x^2 - 5)(x + 5) + 25\).

Q13. Find the product: \((5 - 2x)(3 + x)\).

Q14. Simplify: \((t + s^2)(t^2 - s)\).

Q15. Simplify: \((a + b + c)(a + b - c)\).

Medium Questions

Q1. Add: \(2p^2 q^2 - 3pq + 4, 5 + 7pq - 3p^2 q^2\).

Q2. Subtract: \(3xy + 5yz - 7zx\) from \(5xy - 2yz - 2zx + 10xyz\).

Q3. Find the areas for: \(10m, 5n\).

Q4. Obtain the product: \(xy, yz, zx\).

Q5. Carry out multiplication: \(ab, a - b\).

Q6. Simplify \(3x(4x - 5) + 3\) and evaluate for \(x = 3\).

Q7. Add: \(2x(z - x - y)\) and \(2y(z - y - x)\).

Q8. Multiply the binomials: \((y - 8)\) and \((3y - 4)\).

Q9. Simplify: \((a^2 + 5)(b^3 + 3) + 5\).

Q10. Simplify: \((a + b)(c - d) + (a - b)(c + d) + 2(ac + bd)\).

Q11. Find the product: \((x + 7y)(7x - y)\).

Q12. Simplify: \((x + y)(x^2 - xy + y^2)\).

Q13. Simplify: \((1.5x - 4y)(1.5x + 4y + 3) - 4.5x + 12y\).

Q14. Carry out multiplication: \(a + b, 7a^2 b^2\).

Q15. Subtract: \(3a(a + b + c) - 2b(a - b + c)\) from \(4c(-a + b + c)\).

Long Questions

Q1. Add: \(l^2 + m^2, m^2 + n^2, n^2 + l^2, 2lm + 2mn + 2nl\).

Q2. Subtract: \(4p^2 q - 3pq + 5pq^2 - 8p + 7q - 10\) from \(18 - 3p - 11q + 5pq - 2pq^2 + 5p^2 q\).

Q3. Complete the table of products for monomials like \(2x, -5y, 3x^2, -4xy, 7x^2 y, -9x^2 y^2\).

Q4. Obtain the volume of rectangular boxes: \(2p, 4q, 8r\).

Q5. Simplify \(a(a^2 + a + 1) + 5\) and find values for \(a=0,1,-1\).

Q6. Multiply the binomials: \((2.5l - 0.5m)\) and \((2.5l + 0.5m)\).

Q7. Find the product: \((a + 3b)\) and \((x + 5)\).

Q8. Simplify: \((x + y)(2x + y) + (x + 2y)(x - y)\).

Q9. Carry out multiplication: \(pq + qr + rp, 0\).

Q10. Complete the table for products like \(x + y - 5, 5xy\).

Q11. Find the product: \(x \times x^2 \times x^3 \times x^4\).