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 $$4(a^2 – \frac { 2 }{ 3 } b^2)$$

(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)$$(p^2 – q^2)(2p + q)$$

(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)$$(a + b + c) (a + b – c)$$

(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: $$(a + b + c) (a + b – c)$$

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

$$= a^2 + ab – ac + ab + b^2 – bc + ac + bc – c^2$$

$$= a^2 + ab + ab – bc + bc – ac + ac + b^2 – c^2$$

$$= a^2 + 2ab + b^2 – c^2 + 0 + 0$$

$$= a^2 + 2ab + b^2 – c^2$$