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.
