NCERT solutions for Exercise 9.4 Class 8 Maths Chapter 9 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) \((2pq + 3q^2)\) and \((3pq – 2q^2)\)

(vi) \((\frac { 3 }{ 4 }a^2 + 3b^2)\) and \(4(a^2 – \frac { 2 }{ 3 } b^2)\)


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Answer :

(i)Multiplying the given expressions as:\( (2x + 5) \times (4x – 3)\)

\(= 2x \times (4x – 3) + 5 \times (4x – 3)\)

\(= (2x \times 4x) – (3 \times 2x) + (5 \times 4x) – (5 \times 3)\)

\(= 8x^2 – 6x + 20x – 15\)

\(= 8x^2 + 14x – 15\)

(ii)Multiplying the given expressions as: \((y – 8) \times (3y – 4)\)

\(= y \times (3y – 4) – 8 \times (3y – 4)\)

\(= (y \times 3y) – (y \times 4) – (8 \times 3y) + (-8 \times -4)\)

\(= 3y^2 – 4y – 24y + 32\)

\(= 3y^2 – 28y + 32\)

(iii)Multiplying the given expressions as: \((2.5l – 0.5m) × (2.5l + 0.5m)\)

\(= (2.5l \times 2.5l) + (2.5l \times 0.5m) – (0.5m \times 2.5l) – (0.5m \times 0.5m)\)

\(= 6.25l^2 + 1.25ml – 1.25ml – 0.25m^2\)

\(= 6.25l^2 + 0 – 0.25m^2\)

\(= 6.25l^2 – 0.25m^2\)

(iv)Multiplying the given expressions as: \((a + 3b) \times (x + 5)\)

\(= a \times (x + 5) + 36 \times (x + 5)\)

\(= (a \times x) + (a \times 5) + (36 \times x) + (36 \times 5)\)

\(= ax + 5a + 3bx + 15b\)

(v) Multiplying the given expressions as:\((2pq + 3q^2) \times (3pq – 2q^2)\)

\(= 2pq \times (3pq – 2q^2) + 3q^2 (3pq – 2q^2)\)

\(= (2pq \times 3pq) – (2pq \times 2q^2) + (3q^2 × 3pq) – (3q^2 \times 2q^2)\)

\(= 6p^2q^2 – 4pq^3 + 9pq^3 – 6q^4\)

\(= 6p^2q^2 + 5pq^3 – 6q^4\)

(vi)Multiplying the given expressions as:\((\frac { 3 }{ 4 }a^2 + 3b^2) \times 4(a^2 – \frac { 2 }{ 3 } b^2)\)

\(=(\frac{3}4a^2+3b^2)\times(4a^2-\frac{8}3b^2)\)

\(=\frac{3}4a^2\times (4a^2-\frac{8}3b^2)+3b^2\times(4a^2-\frac{8}3b^2)\)

\(=(\frac{3}4a^2\times 4b^2) - (\frac{3}4a^2\times \frac{8}3b^2) + (3b^2\times 4a^2)-(3b^2\times\frac{8}3b^2)\)

\(=3a^4-2a^2b^2+12a^2b^2-8b^4\)

\(=3a^4+10a^2b^2-8b^4\)

2.Find the product:
(i) \((5 – 2x) (3 + x)\)

(ii)\( (x + 7y) (7x – y)\)

(iii) \((a^2 + b) (a + b^2)\)

(iv)\( (p^2 – q^2)(2p + q)\)


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Answer :

(i)Product of the expression:\( (5 – 2x) (3 + x)\)

\(= 5(3 + x) – 2x(3 + x)\)

\(= (5 \times 3) + (5 \times x) – (2x \times 3) – (2x \times x)\)

\(= 15 + 5x – 6x – 2x^2\)

(ii)Product of the expression: \((x + 7y) (7x – y)\)

\(= x(7x – y) + 7y(7x – y)\)

\(= (x \times 7x) – (x \times y) + (7y \times 7x) – (7y \times y)\)

\(= 7x^2 – xy + 49xy – 7y^2\)

\(= 7x^2 + 48xy – 7y^2\)

(iii)Product of the expression: \((a^2 + b) (a + b^2)\)

\(= a^2 (a + b^2) + b(a + b^2)\)

\(= (a^2 \times a) + (a^2 \times b^2) + (b \times a) + (b \times b^2)\)

\(= a^3 + a^2b^2 + ab + b^3\)

(iv)Product of the expression:\( (p^2 – q^2)(2p + q)\)

\(= p^2(2p + q) – q^2(2p + q)\)

\(= (p^2 \times 2p) + (p^2 \times q) – (q^2 \times 2p) – (q^2 \times q)\)

\(= 2p^3 + p^2q – 2pq^2 – q^3\)

3.Simplify: (i) \((x^2 – 5) (x + 5) + 25\)

(ii) \((a^2 + 5)(b^3 + 3) + 5\)

(iii) \((t + s^2) (t^2 – s)\)

(iv)\( (a + b) (c – d) + (a – b) (c + d) + 2(ac + bd)\)

(v)\( (x + y) (2x + y) + (x + 2y) (x – y)\)

(vi) \((x + y)(x^2 – xy + y^2)\)

(vii)\( (1.5x – 4y)(1.5x + 4y + 3) – 4.5x + 12y\)

(viii)\( (a + b + c) (a + b – c)\)


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Answer :

(i)Simplifying the expression, we have: \((x^2 – 5) (x + 5) + 25\)

\(= x^2 \times (x + 5) + 5\times (x + 5) + 25\)

\(= x^3 + 5x^2 – 5x – 25 + 25\)

\(= x^3 + 5x^2 – 5x + 0\)

\(= x^3 + 5x^2 – 5x\)

(ii)Simplifying the expression, we have: \((a^2 + 5)(b^3 + 3) + 5\)

\(= a^2\times (b^3 + 3) + 5\times (b^3 + 3) + 5\)

\(= a^2b^3 + 3a^2 + 5b^3 + 15 + 5\)

\(= a^2b^3 + 3a^2 + 5b^3 + 20\)

(iii)Simplifying the expression, we have:\( (t + s^2) (t^2 – s)\)

\(= t\times (t^2 – s) + s^2 \times (t^2 – s)\)

\(= t^3 – st + s^2t^2 – s^3\)

\(= t^3 + s^2t^2 – st – s^3\)

(iv) Simplifying the expression, we have:\((a + b)(c – d) + (a – b) (c + d) + 2(ac + bd)\)

\(= a\times(c – d) + b\times(c – d) + a\times(c + d) – b\times(c + d) + 2ac + 2bd\)

\(= ac – ad + bc – bd + ac + ad – bc – bd + 2ac + 2bd\)

\(= ac + ac + 2ac + bc – bc – ad + ad – bd – bd + 2bd\)

\(= 4ac + 0 + 0 + 0\)

\(= 4ac\)

(v) Simplifying the expression, we have:\((x + y) (2x + y) + (x + 2y) (x – y)\)

\(= x\times(2x + y) + y\times(2x + y) + x\times(x – y) + 2y\times(x – y)\)

\(= 2x^2 + xy + 2xy + y^2 + x^2 – xy + 2xy – 2y^2\)

\(= 2x^2 + x^2 + xy + 2xy – xy + 2xy + y^2 – 2y^2\)

\(= 3x^2 + 4xy – y^2\)

(vi)Simplifying the expression, we have:\( (x + y)(x^2 – xy + y^2)\)

\(= x\times(x^2 – xy + y^2) + y\times (x^2 – xy + y^2)\)

\(= x^3 – x^2y + x^2y + xy^2 – xy^2 + y^3\)

\(= x^3 – 0 + 0 + y^3\)

\(= x^3 + y^3\)

(vii)Simplifying the expression, we have:\( (1.5x – 4y)(1.5x + 4y + 3) – 4.5x.+ 12y\)

\(= 1.5x \times (1.5x + 4y + 3) – 4y\times(1.5x + 4y + 3) – 4.5x + 12y\)

\(= 2.25x^2 + 6xy + 4.5x – 6xy – 16y^2 – 12y – 4.5x + 12y\)

\(= 2.25x^2 + 6xy – 6xy + 4.5x – 4.5x + 12y – 12y – 16y^2\)

\(= 2.25x^2 + 0 + 0 + 0 – 16y62\)

\(= 2.25x^2 – 16y^2\)

(viii)Simplifying the expression, we have: \((a + b + c) (a + b – c)\)

\(= a(a + b – c) + b(a + b – c) + c(a + b – c)\)

\(= a^2 + ab – ac + ab + b^2 – bc + ac + bc – c^2\)

\(= a^2 + ab + ab – bc + bc – ac + ac + b^2 – c^2\)

\(= a^2 + 2ab + b^2 – c^2 + 0 + 0\)

\(= a^2 + 2ab + b^2 – c^2\)




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