Welcome dear students! Today we are going to learn about Algebraic Expressions and Identities from Class 8 Maths. In earlier classes, we have already become familiar with what algebraic expressions are. Examples are x + 3, 2y – 5, 3x^2, 4xy + 7. We also learnt how to add and subtract them. To add 7x^2 – 4x + 5 and 9x – 10, we write each expression in a separate row, align like terms vertically, and add them. 5 + (–10) = –5, and –4x + 9x = 5x. The sum is 7x^2 + 5x – 5.
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Example 1: Add 7xy + 5yz – 3zx, 4yz + 9zx – 4y, and –3xz + 5x – 2xy. Writing them in rows with like terms aligned: 7xy + 5yz – 3zx, plus 4yz + 9zx – 4y, plus –2xy – 3zx + 5x. Note xz is the same as zx. Adding column by column gives 5xy + 9yz + 3zx + 5x – 4y. Terms like –4y and 5x are carried over as they have no like terms elsewhere. Example 2: Subtract 5x^2 – 4y^2 + 6y – 3 from 7x^2 – 4xy + 8y^2 + 5x – 3y. We write the first expression, then the second below it. For subtraction, we change the signs of the second expression: –5x^2 becomes +5x^2, –(–4y^2) becomes +4y^2, –6y becomes –6y, and –(–3) becomes +3. Adding vertically: 2x^2 – 4xy + 12y^2 + 5x – 9y + 3. Remember, subtracting a number is adding its additive inverse.
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Now, Exercise 8.1. Question 1: Add the following. (i) ab – bc, bc – ca, ca – ab. Adding like terms: ab – ab = 0, –bc + bc = 0, –ca + ca = 0. Sum is 0. (ii) a – b + ab, b – c + bc, c – a + ac. a – a = 0, –b + b = 0, –c + c = 0. Sum is ab + bc + ac. (iii) 2p^2q^2 – 3pq + 4, and 5 + 7pq – 3p^2q^2. 2p^2q^2 – 3p^2q^2 = –p^2q^2. –3pq + 7pq = 4pq. 4 + 5 = 9. Sum is –p^2q^2 + 4pq + 9. (iv) l^2 + m^2, m^2 + n^2, n^2 + l^2, 2lm + 2mn + 2nl. l^2 + l^2 = 2l^2. m^2 + m^2 = 2m^2. n^2 + n^2 = 2n^2. Sum is 2l^2 + 2m^2 + 2n^2 + 2lm + 2mn + 2nl.
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Question 2: (a) Subtract 4a – 7ab + 3b + 12 from 12a – 9ab + 5b – 3. Change signs of subtrahend: –4a + 7ab – 3b – 12. Combine: 8a – 2ab + 2b – 15. (b) Subtract 3xy + 5yz – 7zx from 5xy – 2yz – 2zx + 10xyz. Change signs: –3xy – 5yz + 7zx. Combine: 2xy – 7yz + 5zx + 10xyz. (c) Subtract 4p^2q – 3pq + 5pq^2 – 8p + 7q – 10 from 18 – 3p – 11q + 5pq – 2pq^2 + 5p^2q. Change signs: –4p^2q + 3pq – 5pq^2 + 8p – 7q + 10. Combine: p^2q + 8pq – 7pq^2 + 5p – 18q + 28.
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Section 8.2: Multiplication of Algebraic Expressions. Consider dot patterns. 4 rows of 9 dots give 4 × 9. 5 rows of 7 dots give 5 × 7. If rows are m + 2 and columns are n + 3, total dots are (m + 2) × (n + 3). Think of area of a rectangle: length l, breadth b, area is l × b. If length becomes l + 5 and breadth becomes b – 3, new area is (l + 5) × (b – 3). Volume of a box is length × breadth × height. Buying items: price per dozen is ₹ p, quantity is z dozens, cost is ₹ p × z. If price is ₹ (p – 2) and quantity is (z – 4) dozens, cost is ₹ (p – 2) × (z – 4). Try These: Think of speed × time, or principal × rate × time for simple interest. We need to multiply algebraic expressions systematically.
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Section 8.3: Multiplying a Monomial by a Monomial. A monomial has one term. 8.3.1: Multiplying two monomials. 4 × x = 4x. 4 × (3x) = 12x. (i) x × 3y = 3xy. (ii) 5x × 3y = 15xy. (iii) 5x × (–3y) = –15xy. (iv) 5x × 4x^2 = 20x^3. (v) 5x × (–4xyz) = –20x^2yz. Coefficient of product = coefficient of first × coefficient of second. Algebraic factor of product = algebraic factor of first × algebraic factor of second. 8.3.2: Multiplying three or more monomials. (i) 2x × 5y × 7z = 10xy × 7z = 70xyz. (ii) 4xy × 5x^2y^2 × 6x^3y^3 = 20x^3y^3 × 6x^3y^3 = 120x^6y^6. Multiply first two, then result by third. Try These: 4x × 5y × 7z. Order does not matter. Result is 140xyz. Also, 4xy × 5x^2y^2 × 6x^3y^3 = (4 × 5 × 6) × (x × x^2 × x^3) × (y × y^2 × y^3) = 120x^6y^6.
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Example 3: Area of rectangle = length × breadth. For length 3x and breadth 5y, the area is 15xy. For 9y and 4y^2, the area is 36y^3. For 4ab and 5bc, the area is 20ab^2c. For 2l^2m and 3lm^2, the area is 6l^3m^3. Example 4: Volume = length × breadth × height. (i) 2ax × 3by × 5cz = 30abcxyz. (ii) m^2n × n^2p × p^2m = m^3n^3p^3. (iii) 2q × 4q^2 × 8q^3 = 64q^6.
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Exercise 8.2. 1. Products: (i) 4 × 7p = 28p. (ii) –4p × 7p = –28p^2. (iii) –4p × 7pq = –28p^2q. (iv) 4p^3 × –3p = –12p^4. (v) 4p × 0 = 0. 2. Areas: (p, q) gives pq. (10m, 5n) gives 50mn. (20x^2, 5y^2) gives 100x^2y^2. (4x, 3x^2) gives 12x^3. (3mn, 4np) gives 12mn^2p. 3. Table of products: Row one, multiplying 2x by each column header gives: 4x^2, –10xy, 6x^3, –8x^2y, 14x^3y, and –18x^3y^2. Row two, multiplying –5y gives: –10xy, 25y^2, –15x^2y, 20xy^2, –35x^2y^2, and 45x^2y^3. Row three, multiplying 3x^2 gives: 6x^3, –15x^2y, 9x^4, –12x^3y, 21x^4y, and –27x^4y^2. Row four, multiplying –4xy gives: –8x^2y, 20xy^2, –12x^3y, 16x^2y^2, –28x^3y^2, and 36x^3y^3. Row five, multiplying 7x^2y gives: 14x^3y, –35x^2y^2, 21x^4y, –28x^3y^2, 49x^4y^2, and –63x^4y^3. Row six, multiplying –9x^2y^2 gives: –18x^3y^2, 45x^2y^3, –27x^4y^2, 36x^3y^3, –63x^4y^3, and 81x^4y^4.
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4. Volumes: (i) 5a × 3a^2 × 7a^4 = 105a^7. (ii) 2p × 4q × 8r = 64pqr. (iii) xy × 2x^2y × 2xy^2 = 4x^4y^4. (iv) a × 2b × 3c = 6abc. 5. Products: (i) xy × yz × zx = x^2y^2z^2. (ii) a × –a^2 × a^3 = –a^6. (iii) 2 × 4y × 8y^2 × 16y^3 = 1024y^6. (iv) a × 2b × 3c × 6abc = 36a^2b^2c^2. (v) m × –mn × mnp = –m^3n^2p.
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Section 8.4: Multiplying a Monomial by a Polynomial. Binomial: two terms. Trinomial: three terms. Polynomial: one or more terms with non-zero coefficients and non-negative integer exponents. 8.4.1: Monomial × Binomial. 3x × (5y + 2) = (3x × 5y) + (3x × 2) = 15xy + 6x. Product is a binomial. Using distributive law: 7 × 106 = 7 × (100 + 6) = 700 + 42 = 742. (–3x) × (–5y + 2) = 15xy – 6x. 5xy × (y^2 + 3) = 5xy^3 + 15xy. Binomial × monomial uses commutative law: (5y + 2) × 3x = 15xy + 6x. Try These: (i) 2x(3x + 5xy) = 6x^2 + 10x^2y. (ii) a^2(2ab – 5c) = 2a^3b – 5a^2c. 8.4.2: Monomial × Trinomial. 3p × (4p^2 + 5p + 7) = 12p^3 + 15p^2 + 21p. Multiply each term and add. Try These: (4p^2 + 5p + 7) × 3p = 12p^3 + 15p^2 + 21p.
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Example 5: Simplify and evaluate. (i) x(x – 3) + 2 = x^2 – 3x + 2. For x = 1: 1 – 3 + 2 = 0. (ii) 3y(2y – 7) – 3(y – 4) – 63 = 6y^2 – 21y – 3y + 12 – 63 = 6y^2 – 24y – 51. For y = –2: 6(4) + 48 – 51 = 24 + 48 – 51 = 21. Example 6: Add. (i) 5m(3 – m) = 15m – 5m^2. Add 6m^2 – 13m: 15m – 5m^2 + 6m^2 – 13m = m^2 + 2m. (ii) 4y(3y^2 + 5y – 7) = 12y^3 + 20y^2 – 28y. 2(y^3 – 4y^2 + 5) = 2y^3 – 8y^2 + 10. Sum: 14y^3 + 12y^2 – 28y + 10. Example 7: Subtract 3pq(p – q) from 2pq(p + q). 3pq(p – q) = 3p^2q – 3pq^2. 2pq(p + q) = 2p^2q + 2pq^2. Subtracting: (2p^2q + 2pq^2) – (3p^2q – 3pq^2) = –p^2q + 5pq^2.
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Exercise 8.3. 1. Products: (i) 4p(q + r) = 4pq + 4pr. (ii) ab(a – b) = a^2b – ab^2. (iii) (a + b)7a^2b^2 = 7a^3b^2 + 7a^2b^3. (iv) (a^2 – 9)4a = 4a^3 – 36a. (v) (pq + qr + rp)0 = 0. 2. Table: (i) a(b + c + d) = ab + ac + ad. (ii) (x + y – 5)5xy = 5x^2y + 5xy^2 – 25xy. (iii) p(6p^2 – 7p + 5) = 6p^3 – 7p^2 + 5p. (iv) 4p^2q^2(p^2 – q^2) = 4p^4q^2 – 4p^2q^4. (v) (a + b + c)abc = a^2bc + ab^2c + abc^2. 3. Products: (i) a^2 × 2a^22 × 4a^26 = 8a^50. (ii) (2/3 xy) × (–9/10 x^2y^2) = –3/5 x^3y^3. (iii) (–10/3 pq^3) × (6/5 p^3q) = –4p^4q^4. (iv) x × x^2 × x^3 × x^4 = x^10.
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4. (a) 3x(4x – 5) + 3 = 12x^2 – 15x + 3. For x = 3: 108 – 45 + 3 = 66. For x = 1/2: 3 – 7.5 + 3 = –1.5. (b) a(a^2 + a + 1) + 5 = a^3 + a^2 + a + 5. For a = 0: 5. For a = 1: 8. For a = –1: –1 + 1 – 1 + 5 = 4. 5. (a) p(p – q) + q(q – r) + r(r – p) = p^2 – pq + q^2 – qr + r^2 – rp. (b) 2x(z – x – y) + 2y(z – y – x) = 2xz – 2x^2 – 2xy + 2yz – 2y^2 – 2xy = –2x^2 – 2y^2 – 4xy + 2xz + 2yz. (c) 4l(10n – 3m + 2l) – 3l(l – 4m + 5n) = 40ln – 12lm + 8l^2 – 3l^2 + 12lm – 15ln = 5l^2 + 25ln. (d) 4c(–a + b + c) – [3a(a + b + c) – 2b(a – b + c)] = –4ac + 4bc + 4c^2 – [3a^2 + 3ab + 3ac – 2ab + 2b^2 – 2bc] = –4ac + 4bc + 4c^2 – 3a^2 – ab – 3ac – 2b^2 + 2bc = –3a^2 – 2b^2 – ab – 7ac + 6bc + 4c^2.
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Section 8.5: Multiplying a Polynomial by a Polynomial. 8.5.1: Binomial × Binomial. (3a + 4b)(2a + 3b) = 3a(2a + 3b) + 4b(2a + 3b) = 6a^2 + 9ab + 8ab + 12b^2 = 6a^2 + 17ab + 12b^2. Combine like terms. Example 8: (i) (x – 4)(2x + 3) = 2x^2 + 3x – 8x – 12 = 2x^2 – 5x – 12. (ii) (x – y)(3x + 5y) = 3x^2 + 5xy – 3xy – 5y^2 = 3x^2 + 2xy – 5y^2. Example 9: (i) (a + 7)(b – 5) = ab – 5a + 7b – 35. No like terms. (ii) (a^2 + 2b^2)(5a – 3b) = 5a^3 – 3a^2b + 10ab^2 – 6b^3. 8.5.2: Binomial × Trinomial. (a + 7)(a^2 + 3a + 5) = a(a^2 + 3a + 5) + 7(a^2 + 3a + 5) = a^3 + 3a^2 + 5a + 7a^2 + 21a + 35 = a^3 + 10a^2 + 26a + 35.
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Example 10: Simplify (a + b)(2a – 3b + c) – (2a – 3b)c. First part: 2a^2 – 3ab + ac + 2ab – 3b^2 + bc = 2a^2 – ab – 3b^2 + bc + ac. Second part: 2ac – 3bc. Subtracting: 2a^2 – ab – 3b^2 + bc + ac – 2ac + 3bc = 2a^2 – ab – 3b^2 + 4bc – ac. Exercise 8.4. 1. (i) (2x + 5)(4x – 3) = 8x^2 + 14x – 15. (ii) (y – 8)(3y – 4) = 3y^2 – 28y + 32. (iii) (2.5l – 0.5m)(2.5l + 0.5m) = 6.25l^2 – 0.25m^2. (iv) (a + 3b)(x + 5) = ax + 5a + 3bx + 15b. (v) (2pq + 3q^2)(3pq – 2q^2) = 6p^2q^2 + 5pq^3 – 6q^4. (vi) (3/4 a^2 + 3b^2)4(a^2 – 2/3 b^2) = 3a^4 + 10a^2b^2 – 8b^4. 2. (i) (5 – 2x)(3 + x) = 15 – x – 2x^2. (ii) (x + 7y)(7x – y) = 7x^2 + 48xy – 7y^2. (iii) (a^2 + b)(a + b^2) = a^3 + a^2b^2 + ab + b^3. (iv) (p^2 – q^2)(2p + q) = 2p^3 + p^2q – 2pq^2 – q^3.
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3. (i) (x^2 – 5)(x + 5) + 25 = x^3 + 5x^2 – 5x – 25 + 25 = x^3 + 5x^2 – 5x. (ii) (a^2 + 5)(b^3 + 3) + 5 = a^2b^3 + 3a^2 + 5b^3 + 15 + 5 = a^2b^3 + 3a^2 + 5b^3 + 20. (iii) (t + s^2)(t^2 – s) = t^3 – st + s^2t^2 – s^3. (iv) (a + b)(c – d) + (a – b)(c + d) + 2(ac + bd) = ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd = 4ac. (v) (x + y)(2x + y) + (x + 2y)(x – y) = 2x^2 + 3xy + y^2 + x^2 + xy – 2y^2 = 3x^2 + 4xy – y^2. (vi) (x + y)(x^2 – xy + y^2) = x^3 + y^3. (vii) (1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y = 2.25x^2 – 16y^2. (viii) (a + b + c)(a + b – c) = (a + b)^2 – c^2 = a^2 + 2ab + b^2 – c^2.
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Summary: Expressions are formed from variables and constants. Terms are added to form expressions. Terms themselves are products of factors. Expressions with one, two, three terms are monomials, binomials, trinomials. Polynomials have one or more terms with non-zero coefficients and non-negative integer exponents. Like terms have same variables and powers. Add or subtract by combining like terms. Multiplication is needed for area, volume, cost. Monomial × monomial = monomial. Multiply polynomial by monomial term by term. Multiply polynomial by binomial or trinomial by multiplying every term of one by every term of the other, then combining like terms.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]