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Class 8 Exam > Class 8 Notes > RD Sharma Solutions for Class 8 Mathematics > RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 )

RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics PDF Download

Question 1: Multiply:
(5x + 3) by (7x + 2)

Answer 1: To multiply, we will use distributive law as follows:
$(5 x + 3) (7 x + 2) = 5 x (7 x + 2) + 3 (7 x + 2) = (5 x \times 7 x + 5 x \times 2) + (3 \times 7 x + 3 \times 2) = (35 x^{2} + 10 x) + (21 x + 6) = 35 x^{2} + 10 x + 21 x + 6 = 35 x^{2} + 31 x + 6$ Thus, the answer is $35 x^{2} + 31 x + 6$ .

Question 2: Multiply:
(2x + 8) by (x − 3)

Answer 2: To multiply the expressions, we will use the distributive law in the following way:
$(2 x + 8) (x - 3) = 2 x (x - 3) + 8 (x - 3) = (2 x \times x - 2 x \times 3) + (8 x - 8 \times 3) = (2 x^{2} - 6 x) + (8 x - 24) = 2 x^{2} - 6 x + 8 x - 24 = 2 x^{2} + 2 x - 24$ Thus, the answer is $2 x^{2} + 2 x - 24$ .

Question 3: Multiply:
(7x + y) by (x + 5y)

Answer 3: To multiply, we will use distributive law as follows:
$(7 x + y) (x + 5 y) = 7 x (x + 5 y) + y (x + 5 y) = 7 x^{2} + 35 x y + x y + 5 y^{2} = 7 x^{2} + 36 x y + 5 y^{2}$
Thus, the answer is $7 x^{2} + 36 x y + 5 y^{2}$ .

Question 4: Multiply:
(a − 1) by (0.1a² + 3)

Answer 4: To multiply, we will use distributive law as follows:
$(a - 1) (0.1 a^{2} + 3) = 0.1 a^{2} (a - 1) + 3 (a - 1) = 0.1 a^{3} - 0.1 a^{2} + 3 a - 3$
Thus, the answer is $0.1 a^{3} - 0.1 a^{2} + 3 a - 3$ .

Question 5: Multiply:
(3x² + y²) by (2x² + 3y²)

Answer 5: To multiply, we will use distributive law as follows:
$(3 x^{2} + y^{2}) (2 x^{2} + 3 y^{2}) = 3 x^{2} (2 x^{2} + 3 y^{2}) + y^{2} (2 x^{2} + 3 y^{2}) = 6 x^{4} + 9 x^{2} y^{2} + 2 x^{2} y^{2} + 3 y^{4} = 6 x^{4} + 11 x^{2} y^{2} + 3 y^{4}$ Thus, the answer is $6 x^{4} + 11 x^{2} y^{2} + 3 y^{4}$ .

Question 6: Multiply:

Answer 6: To multiply, we will use distributive law as follows:

Thus, the answer is

Question 7: Multiply:
(x⁶ − y⁶) by (x²+ y²)

Answer 7: To multiply, we will use distributive law as follows:

(x6−y6)(x2+y2)=x6(x2+y2)−y6(x2+y2)=(x8+x6y2)−(y6x2+y8)=x8+x6y2−y6x2−y8 Thus, the answer is $x^{8} + x^{6} y^{2} - y^{6} x^{2} - y^{8}$ .

Question 8: Multiply:
(x² + y²) by (3a + 2b)

Answer 8: To multiply, we will use distributive law as follows:

(x2+y2)(3a+2b)=x2(3a+2b)+y2(3a+2b)=3ax2+2bx2+3ay2+2by2 Thus, the answer is $3 a x^{2} + 2 b x^{2} + 3 a y^{2} + 2 b y^{2}$ .

Question 9: Multiply:
[−3d + (−7f)] by (5d + f)

Answer 9: To multiply, we will use distributive law as follows:
$[- 3 d + (- 7 f)] (5 d + f) = (- 3 d) (5 d + f) + (- 7 f) (5 d + f) = (- 15 d^{2} - 3 d f) + (- 35 d f - 7 f^{2}) = - 15 d^{2} - 3 d f - 35 d f - 7 f^{2} = - 15 d^{2} - 38 d f - 7 f^{2}$ Thus, the answer is $- 15 d^{2} - 38 d f - 7 f^{2}$ .

Question 10: Multiply:
(0.8a − 0.5b) by (1.5a − 3b)

Answer 10: To multiply, we will use distributive law as follows:
$(0.8 a - 0.5 b) (1.5 a - 3 b) = 0.8 a (1.5 a - 3 b) - 0.5 b (1.5 a - 3 b) = 1.2 a^{2} - 2.4 a b - 0.75 a b + 1.5 b^{2} = 1.2 a^{2} - 3.15 a b + 1.5 b^{2}$ Thus, the answer is $1.2 a^{2} - 3.15 a b + 1.5 b^{2}$ .

Question 11: Multiply:
(2x²y² − 5xy²) by (x² − y²)

Answer 11: To multiply, we will use distributive law as follows:

(2x2y2−5xy2)(x2−y2)=2x2y2(x2−y2)−5xy2(x2−y2)=2x4y2−2x2y4−5x3y2+5xy4 Thus, the answer is $2 x^{4} y^{2} - 2 x^{2} y^{4} - 5 x^{3} y^{2} + 5 x y^{4}$ .

Question 12: Multiply:

Answer 12: To multiply the expressions, we will use the distributive law in the following way:

Thus, the answer is

Question 13: Multiply:

Answer 13: To multiply, we will use distributive law as follows:

RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

Thus, the answer is RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

Question 14: Multiply:

(3x²y − 5xy²) by

Answer 14: To multiply, we will use distributive law as follows:

(3x2y−5xy2) (1/5 x2+1/3 y2)

= 1/5 x2(3x2y−5xy2)+1/3 y2(3x2y−5xy2)

= 3/5 x4y−x3y2+x2y3−5/3 xy4

Thus, the answer is 3/5 x4y−x3y2+x2y3−5/3 xy4

Question 15: Multiply:
(2x² − 1) by (4x³ + 5x²)

Answer 15: To multiply, we will use distributive law as follows:
$(2 x^{2} - 1) (4 x^{3} + 5 x^{2}) = 2 x^{2} (4 x^{3} + 5 x^{2}) - 1 (4 x^{3} + 5 x^{2}) = 8 x^{5} + 10 x^{4} - 4 x^{3} - 5 x^{2}$ Thus, the answer is $8 x^{5} + 10 x^{4} - 4 x^{3} - 5 x^{2}$ .

Question 16: (2xy + 3y²) (3y² − 2)

Answer 16: To multiply, we will use distributive law as follows:
$(2 x y + 3 y^{2}) (3 y^{2} - 2) = 2 x y (3 y^{2} - 2) + 3 y^{2} (3 y^{2} - 2) = 6 x y^{3} - 4 x y + 9 y^{4} - 6 y^{2} = 9 y^{4} + 6 x y^{3} - 6 y^{2} - 4 x y$ Thus, the answer is $9 y^{4} + 6 x y^{3} - 6 y^{2} - 4 x y$ .

Question 17: Find the following product and verify the result for x = − 1, y = − 2:
(3x − 5y) (x + y)

Answer 17: To multiply, we will use distributive law as follows:
$(3 x - 5 y) (x + y) = 3 x (x + y) - 5 y (x + y) = 3 x^{2} + 3 x y - 5 x y - 5 y^{2} = 3 x^{2} - 2 x y - 5 y^{2}$ $∴$ $(3 x - 5 y) (x + y) = 3 x^{2} - 2 x y - 5 y^{2}$ .

Now, we put x = $-$ 1 and y = $-$ 2 on both sides to verify the result.

LHS=(3x−5y)(x+y)={3(−1)−5(−2)}{−1+(−2)}=(−3+10)(−3)=(7)(−3)=−21 RHS=3x2−2xy−5y2=3(−1)2−2(−1)(−2)−5(−2)2=3×1−4−5×4=3−4−20=−21 Because LHS is equal to RHS, the result is verified.
Thus, the answer is $3 x^{2} - 2 x y - 5 y^{2}$ .

Question 18: Find the following product and verify the result for x = − 1, y = − 2:
(x²y − 1) (3 − 2x²y)

Answer 18: To multiply, we will use distributive law as follows:

(x2y−1)(3−2x2y)=x2y(3−2x2y)−1×(3−2x2y)=3x2y−2x4y2−3+2x2y=5x2y−2x4y2−3 $∴$ $(x^{2} y - 1) (3 - 2 x^{2} y) = 5 x^{2} y - 2 x^{4} y^{2} - 3$

Now, we put x = $-$ 1 and y = $-$ 2 on both sides to verify the result.

LHS = (x2y−1)(3−2x2y)=[(−1)2(−2)−1][3−2(−1)2(−2)]=[1×(−2)−1][3−2×1×(−2)]=(−2−1)(3+4)=−3×7=−21RHS=5x2y−2x4y2−3=5(−1)2(−2)−2(−1)4(−2)2−3=[5×1×(−2)]−[2×1×4]−3=−10−8−3=−21 Because LHS is equal to RHS, the result is verified.
Thus, the answer is $5 x^{2} y - 2 x^{4} y^{2} - 3$ .

Question 19: Find the following product and verify the result for x = − 1, y = − 2:

Answer 19: To multiply, we will use distributive law as follows:

RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

Now, we will put x = $-$ 1 and y = $-$ 2 on both the sides to verify the result.

RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

=- 119/225

RHS= RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

= - 119/225

Because LHS is equal to RHS, the result is verified.

Thus, the answer is RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics

Question 20: Simplify:
x²(x + 2y) (x − 3y)

Answer 20: To simplify, we will proceed as follows:

x2(x+2y)(x−3y)=[x2(x+2y)](x−3y)=(x3+2x2y)(x−3y)=x3(x−3y)+2x2y(x−3y)=x4−3x3y+2x3y−6x2y2=x4−x3y−6x2y2 Thus, the answer is $x^{4} - x^{3} y - 6 x^{2} y^{2}$ .

Question 21: Simplify:
(x² − 2y²) (x + 4y) x²y²

Answer 21: To simplify, we will proceed as follows:

(x2−2y2)(x+4y)x2y2=[x2(x+4y)−2y2(x+4y)]x2y2=(x3+4x2y−2xy2−8y3)x2y2=x5y2+4x4y3−2x3y4−8x2y5 Thus, the answer is $x^{5} y^{2} + 4 x^{4} y^{3} - 2 x^{3} y^{4} - 8 x^{2} y^{5}$ .

Question 22: Simplify:
a²b²(a + 2b)(3a + b)

Answer 22: To simplify, we will proceed as follows:

a2b2(a+2b)(3a+b)=[a2b2(a+2b)](3a+b)=(a3b2+2a2b3)(3a+b)=3a(a3b2+2a2b3)+b(a3b2+2a2b3)=3a4b2+6a3b3+a3b3+2a2b4=3a4b2+7a3b3+2a2b4 Thus, the answer is $3 a^{4} b^{2} + 7 a^{3} b^{3} + 2 a^{2} b^{4}$ .

Question 23: Simplify:
x²(x − y) y²(x + 2y)

Answer 23: To simplify, we will proceed as follows:

x2(x−y)y2(x+2y)=[x2(x−y)][y2(x+2y)]=(x3−x2y)(xy2+2y3)=x3(xy2+2y3)−x2y(xy2+2y3)=x4y2+2x3y3−[x3y3+2x2y4]=x4y2+2x3y3−x3y3−2x2y4=x4y2+x3y3−2x2y4 Thus, the answer is $x^{4} y^{2} + x^{3} y^{3} - 2 x^{2} y^{4}$ .

Question 24: Simplify:
(x³ − 2x² + 5x − 7)(2x − 3)

Answer 24: To simplify, we will proceed as follows:

(x3−2x2+5x−7)(2x−3)=2x(x3−2x2+5x−7)−3(x3−2x2+5x−7)=2x4−4x3+10x2−14x−3x3+6x2−15x+21 $= 2 x^{4} - 4 x^{3} - 3 x^{3} + 10 x^{2} + 6 x^{2} - 14 x - 15 x + 21$ (Rearranging)
$= 2 x^{4} - 7 x^{3} + 16 x^{2} - 29 x + 21$ (Combining like terms)

Thus, the answer is $2 x^{4} - 7 x^{3} + 16 x^{2} - 29 x + 21$ .

Question 25: Simplify:
(5x + 3)(x − 1)(3x − 2)

Answer 25: To simplify, we will proceed as follows:
$(5 x + 3) (x - 1) (3 x - 2) = [(5 x + 3) (x - 1)] (3 x - 2)$ $= [5 x (x - 1) + 3 (x - 1)] (3 x - 2)$ (Distributive law)
$= [5 x^{2} - 5 x + 3 x - 3] (3 x - 2) = [5 x^{2} - 2 x - 3] (3 x - 2) = 3 x (5 x^{2} - 2 x - 3) - 2 (5 x^{2} - 2 x - 3) = 15 x^{3} - 6 x^{2} - 9 x - [10 x^{2} - 4 x - 6] = 15 x^{3} - 6 x^{2} - 9 x - 10 x^{2} + 4 x + 6$ $= 15 x^{3} - 6 x^{2} - 10 x^{2} - 9 x + 4 x + 6$ (Rearranging)
$= 15 x^{3} - 16 x^{2} - 5 x + 6$ (Combining like terms)

Thus, the answer is $15 x^{3} - 16 x^{2} - 5 x + 6$ .

Question 26: Simplify:
(5 − x)(6 − 5x)( 2 − x)

Answer 26: To simplify, we will proceed as follows:

(5−x)(6−5x)(2−x)=[(5−x)(6−5x)](2−x) $= [5 (6 - 5 x) - x (6 - 5 x)] (2 - x)$ (Distributive law)
$= (30 - 25 x - 6 x + 5 x^{2}) (2 - x) = (30 - 31 x + 5 x^{2}) (2 - x) = 2 (30 - 31 x + 5 x^{2}) - x (30 - 31 x + 5 x^{2}) = 60 - 62 x + 10 x^{2} - 30 x + 31 x^{2} - 5 x^{3}$ $= 60 - 62 x - 30 x + 10 x^{2} + 31 x^{2} - 5 x^{3}$ (Rearranging)
$= 60 - 92 x + 41 x^{2} - 5 x^{3}$ (Combining like terms)

Thus, the answer is $60 - 92 x + 41 x^{2} - 5 x^{3}$ .

Question 27: Simplify:
(2x² + 3x − 5)(3x² − 5x + 4)

Answer 27: To simplify, we will proceed as follows:

$(2 x^{2} + 3 x - 5) (3 x^{2} - 5 x + 4)$
$= 2 x^{2} (3 x^{2} - 5 x + 4) + 3 x (3 x^{2} - 5 x + 4) - 5 (3 x^{2} - 5 x + 4)$ (Distributive law)
$= 6 x^{4} - 10 x^{3} + 8 x^{2} + 9 x^{3} - 15 x^{2} + 12 x - 15 x^{2} + 25 x - 20$
$= 6 x^{4} - 10 x^{3} + 9 x^{3} + 8 x^{2} - 15 x^{2} - 15 x^{2} + 12 x + 25 x - 20$ (Rearranging)
$= 6 x^{4} - x^{3} - 22 x^{2} + 36 x - 20$ (Combining like terms)

Thus, the answer is $6 x^{4} - x^{3} - 22 x^{2} + 36 x - 20$ .

Question 28: Simplify:
(3x − 2)(2x − 3) + (5x − 3)(x + 1)

Answer 28: To simplify, we will proceed as follows:

(3x−2)(2x−3)+(5x−3)(x+1)=[(3x−2)(2x−3)]+[(5x−3)(x+1)] $= [3 x (2 x - 3) - 2 (2 x - 3)] + [5 x (x + 1) - 3 (x + 1)]$ (Distributive law)
$= 6 x^{2} - 9 x - 4 x + 6 + 5 x^{2} + 5 x - 3 x - 3$
$= 6 x^{2} + 5 x^{2} - 9 x - 4 x + 5 x - 3 x - 3 + 6$ (Rearranging)
$= 11 x^{2} - 11 x + 3$ (Combining like terms)

Thus, the answer is $11 x^{2} - 11 x + 3$ .

Question 29: Simplify:
(5x − 3)(x + 2) − (2x + 5)(4x − 3)

Answer 29: To simplify, we will proceed as follows:

(5x−3)(x+2)−(2x+5)(4x−3)=[(5x−3)(x+2)]−[(2x+5)(4x−3)] $= [5 x (x + 2) - 3 (x + 2)] - [2 x (4 x - 3) + 5 (4 x - 3)]$ (Distributive law)
$= 5 x^{2} + 10 x - 3 x - 6 - 8 x^{2} + 6 x - 20 x + 15$
$= 5 x^{2} - 8 x^{2} + 10 x - 3 x + 6 x - 20 x - 6 + 15$ (Rearranging)
$= 5 x^{2} - 8 x^{2} + 10 x - 3 x + 6 x - 20 x - 6 + 15 = - 3 x^{2} - 7 x + 9$ (Combining like terms)

Hence, the answer is $- 3 x^{2} - 7 x + 9$ .

Question 30: Simplify:
(3x + 2y)(4x + 3y) − (2x − y)(7x − 3y)

Answer 30: To simplify, we will proceed as follows:

(3x+2y)(4x+3y)−(2x−y)(7x−3y)=[(3x+2y)(4x+3y)]−[(2x−y)(7x−3y)] $= [3 x (4 x + 3 y) + 2 y (4 x + 3 y)] - [2 x (7 x - 3 y) - y (7 x - 3 y)]$ (Distributive law)
$= 12 x^{2} + 9 x y + 8 x y + 6 y^{2} - [14 x^{2} - 6 x y - 7 x y + 3 y^{2}] = 12 x^{2} + 9 x y + 8 x y + 6 y^{2} - 14 x^{2} + 6 x y + 7 x y - 3 y^{2}$

$= 12 x^{2} + 9 x y + 8 x y + 6 y^{2} - [14 x^{2} - 6 x y - 7 x y + 3 y^{2}] = 12 x^{2} + 9 x y + 8 x y + 6 y^{2} - 14 x^{2} + 6 x y + 7 x y - 3 y^{2}$

Question 31: Simplify:
(x² − 3x + 2)(5x − 2) − (3x² + 4x − 5)(2x − 1)

Answer 31: To simplify, we will proceed as follows:

$(x^{2} - 3 x + 2) (5 x - 2) - (3 x^{2} + 4 x - 5) (2 x - 1) = [(x^{2} - 3 x + 2) (5 x - 2)] - [(3 x^{2} + 4 x - 5) (2 x - 1)]$
$= [5 x (x^{2} - 3 x + 2) - 2 (x^{2} - 3 x + 2)] - [2 x (3 x^{2} + 4 x - 5) - 1 \times (3 x^{2} + 4 x - 5)]$ (Distributive law)
$= [5 x^{3} - 15 x^{2} + 10 x - (2 x^{2} - 6 x + 4)] - [6 x^{3} + 8 x^{2} - 10 x - 3 x^{2} - 4 x + 5] = [5 x^{3} - 15 x^{2} + 10 x - 2 x^{2} + 6 x - 4] - [6 x^{3} + 8 x^{2} - 10 x - 3 x^{2} - 4 x + 5] = 5 x^{3} - 15 x^{2} + 10 x - 2 x^{2} + 6 x - 4 - 6 x^{3} - 8 x^{2} + 10 x + 3 x^{2} + 4 x - 5$
$= 5 x^{3} - 6 x^{3} - 15 x^{2} - 2 x^{2} - 8 x^{2} + 3 x^{2} + 10 x + 6 x + 10 x + 4 x - 5 - 4$ (Rearranging)
$= - x^{3} - 22 x^{2} + 30 x - 9$ (Combining like terms)

Thus, the answer is $- x^{3} - 22 x^{2} + 30 x - 9$ .

Question 32: Simplify:
(x³ − 2x² + 3x − 4) (x −1) − (2x − 3)(x² − x + 1)

Answer 32: To simplify,we will proceed as follows:

(x3−2x2+3x−4)(x−1)−(2x−3)(x2−x+1)=[(x3−2x2+3x−4)(x−1)]−[(2x−3)(x2−x+1)] $= [x (x^{3} - 2 x^{2} + 3 x - 4) - 1 (x^{3} - 2 x^{2} + 3 x - 4)] - [2 x (x^{2} - x + 1) - 3 (x^{2} - x + 1)]$ (Distributive law)

=[x(x3−2x2+3x−4)−1(x3−2x2+3x−4)]−[2x(x2−x+1)−3(x2−x+1)]=x4−2x3+3x2−4x−x3+2x2−3x+4−[2x3−2x2+2x−3x2+3x−3]=x4−2x3+3x2−4x−x3+2x2−3x+4−2x3+2x2−2x+3x2−3x+3 $= x^{4} - 2 x^{3} - 2 x^{3} - x^{3} + 3 x^{2} + 2 x^{2} + 2 x^{2} + 3 x^{2} - 4 x - 3 x - 2 x - 3 x + 4 + 3$ (Rearranging)
$= x^{4} - 5 x^{3} + 10 x^{2} - 12 x + 7$ (Combining like terms)

Thus, the answer is $x^{4} - 5 x^{3} + 10 x^{2} - 12 x + 7$ .

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RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) Notes

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RD Sharma Solutions for Class 8 Math Chapter 6 - Algebraic Expressions and Identities (Part-4 ) | RD Sharma Solutions for Class 8 Mathematics PDF Download

Question 1: Multiply:(5x + 3) by (7x + 2)

Answer 1: To multiply, we will use distributive law as follows:(5x+3)(7x+2)=5x(7x+2)+3(7x+2)=(5x×7x+5x×2)+(3×7x+3×2)=(35x2+10x)+(21x+6)=35x2+10x+21x+6=35x2+31x+6Thus, the answer is 35x2+31x+635x2+31x+6.

Question 2: Multiply:(2x + 8) by (x − 3)

Question 3: Multiply:(7x + y) by (x + 5y)

Answer 3: To multiply, we will use distributive law as follows:(7x+y)(x+5y)=7x(x+5y)+y(x+5y)=7x2+35xy+xy+5y2=7x2+36xy+5y27x+yx+5y=7xx+5y+yx+5y=7x2+35xy+xy+5y2=7x2+36xy+5y2Thus, the answer is 7x2+36xy+5y27x2+36xy+5y2.

Question 4: Multiply:(a − 1) by (0.1a2 + 3)

Answer 4: To multiply, we will use distributive law as follows:(a−1)(0.1a2+3)=0.1a2(a−1)+3(a−1)=0.1a3−0.1a2+3a−3a-10.1a2+3=0.1a2a-1+3a-1=0.1a3-0.1a2+3a-3Thus, the answer is 0.1a3−0.1a2+3a−30.1a3-0.1a2+3a-3.

Question 5: Multiply:(3x2 + y2) by (2x2 + 3y2)

Answer 5: To multiply, we will use distributive law as follows:(3x2+y2)(2x2+3y2)=3x2(2x2+3y2)+y2(2x2+3y2)=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3y43x2+y22x2+3y2=3x22x2+3y2+y22x2+3y2=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3yThus, the answer is 6x4+11x2y2+3y46x4+11x2y2+3y4.

Question 6: Multiply:

Answer 6: To multiply, we will use distributive law as follows:

Question 7: Multiply:(x6 − y6) by (x2 + y2)

Answer 7: To multiply, we will use distributive law as follows:

Question 8: Multiply:(x2 + y2) by (3a + 2b)

Answer 8: To multiply, we will use distributive law as follows:

Question 9: Multiply:[−3d + (−7f)] by (5d + f)

Answer 9: To multiply, we will use distributive law as follows:[−3d+(−7f)](5d+f)=(−3d)(5d+f)+(−7f)(5d+f)=(−15d2−3df)+(−35df−7f2)=−15d2−3df−35df−7f2=−15d2−38df−7f2Thus, the answer is −15d2−38df−7f2-15d2-38df-7f2.

Question 10: Multiply:(0.8a − 0.5b) by (1.5a − 3b)

Answer 10: To multiply, we will use distributive law as follows:(0.8a−0.5b)(1.5a−3b)=0.8a(1.5a−3b)−0.5b(1.5a−3b)=1.2a2−2.4ab−0.75ab+1.5b2=1.2a2−3.15ab+1.5b2Thus, the answer is 1.2a2−3.15ab+1.5b21.2a2-3.15ab+1.5b2.

Question 11: Multiply:(2x2y2 − 5xy2) by (x2 − y2)

Answer 11: To multiply, we will use distributive law as follows:

(2x2y2−5xy2)(x2−y2)=2x2y2(x2−y2)−5xy2(x2−y2)=2x4y2−2x2y4−5x3y2+5xy4 Thus, the answer is 2x4y2−2x2y4−5x3y2+5xy42x4y2-2x2y4-5x3y2+5xy4.

Question 12: Multiply:

Answer 12: To multiply the expressions, we will use the distributive law in the following way:

Question 13: Multiply:

Answer 13: To multiply, we will use distributive law as follows:

Question 14: Multiply:

Answer 14: To multiply, we will use distributive law as follows:

Question 15: Multiply:(2x2 − 1) by (4x3 + 5x2)

Answer 15: To multiply, we will use distributive law as follows:(2x2−1)(4x3+5x2)=2x2(4x3+5x2)−1(4x3+5x2)=8x5+10x4−4x3−5x2Thus, the answer is 8x5+10x4−4x3−5x28x5+10x4-4x3-5x2.

Question 16: (2xy + 3y2) (3y2 − 2)

Answer 16: To multiply, we will use distributive law as follows:(2xy+3y2)(3y2−2)=2xy(3y2−2)+3y2(3y2−2)=6xy3−4xy+9y4−6y2=9y4+6xy3−6y2−4xyThus, the answer is 9y4+6xy3−6y2−4xy9y4+6xy3-6y2-4xy.

Question 17: Find the following product and verify the result for x = − 1, y = − 2:(3x − 5y) (x + y)

Answer 17: To multiply, we will use distributive law as follows:(3x−5y)(x+y)=3x(x+y)−5y(x+y)=3x2+3xy−5xy−5y2=3x2−2xy−5y2∴∴ (3x−5y)(x+y)=3x2−2xy−5y23x-5yx+y=3x2-2xy-5y2.

Now, we put x = −-1 and y = −-2 on both sides to verify the result.

LHS=(3x−5y)(x+y)={3(−1)−5(−2)}{−1+(−2)}=(−3+10)(−3)=(7)(−3)=−21 RHS=3x2−2xy−5y2=3(−1)2−2(−1)(−2)−5(−2)2=3×1−4−5×4=3−4−20=−21 Because LHS is equal to RHS, the result is verified.Thus, the answer is 3x2−2xy−5y23x2-2xy-5y2.

Question 18: Find the following product and verify the result for x = − 1, y = − 2:(x2y − 1) (3 − 2x2y)

Answer 18: To multiply, we will use distributive law as follows:

Question 19: Find the following product and verify the result for x = − 1, y = − 2:

Answer 19: To multiply, we will use distributive law as follows:

Question 20: Simplify:x2(x + 2y) (x − 3y)

Answer 20: To simplify, we will proceed as follows:

Question 21: Simplify:(x2 − 2y2) (x + 4y) x2y2

Answer 21: To simplify, we will proceed as follows:

Question 22: Simplify:a2b2(a + 2b)(3a + b)

Answer 22: To simplify, we will proceed as follows:

Question 23: Simplify:x2(x − y) y2(x + 2y)

Answer 23: To simplify, we will proceed as follows:

Question 24: Simplify:(x3 − 2x2 + 5x − 7)(2x − 3)

Answer 24: To simplify, we will proceed as follows:

Question 25: Simplify:(5x + 3)(x − 1)(3x − 2)

Question 26: Simplify:(5 − x)(6 − 5x)( 2 − x)

Answer 26: To simplify, we will proceed as follows:

Question 27: Simplify:(2x2 + 3x − 5)(3x2 − 5x + 4)

Answer 27: To simplify, we will proceed as follows:

Question 28: Simplify:(3x − 2)(2x − 3) + (5x − 3)(x + 1)

Answer 28: To simplify, we will proceed as follows:

Question 29: Simplify:(5x − 3)(x + 2) − (2x + 5)(4x − 3)

Answer 29: To simplify, we will proceed as follows:

Question 30: Simplify:(3x + 2y)(4x + 3y) − (2x − y)(7x − 3y)

Answer 30: To simplify, we will proceed as follows:

Question 31: Simplify:(x2 − 3x + 2)(5x − 2) − (3x2 + 4x − 5)(2x − 1)

Answer 31: To simplify, we will proceed as follows:

Question 32: Simplify:(x3 − 2x2 + 3x − 4) (x −1) − (2x − 3)(x2 − x + 1)

Answer 32: To simplify,we will proceed as follows:

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Objective type Questions

Question 1: Multiply:
(5x + 3) by (7x + 2)

Question 2: Multiply:
(2x + 8) by (x − 3)

Question 3: Multiply:
(7x + y) by (x + 5y)

Answer 3: To multiply, we will use distributive law as follows:
$(7 x + y) (x + 5 y) = 7 x (x + 5 y) + y (x + 5 y) = 7 x^{2} + 35 x y + x y + 5 y^{2} = 7 x^{2} + 36 x y + 5 y^{2}$
Thus, the answer is $7 x^{2} + 36 x y + 5 y^{2}$ .

Question 4: Multiply:
(a − 1) by (0.1a² + 3)

Answer 4: To multiply, we will use distributive law as follows:
$(a - 1) (0.1 a^{2} + 3) = 0.1 a^{2} (a - 1) + 3 (a - 1) = 0.1 a^{3} - 0.1 a^{2} + 3 a - 3$
Thus, the answer is $0.1 a^{3} - 0.1 a^{2} + 3 a - 3$ .

Question 5: Multiply:
(3x² + y²) by (2x² + 3y²)

Question 7: Multiply:
(x⁶ − y⁶) by (x²+ y²)

Question 8: Multiply:
(x² + y²) by (3a + 2b)

Question 9: Multiply:
[−3d + (−7f)] by (5d + f)

Question 10: Multiply:
(0.8a − 0.5b) by (1.5a − 3b)

Answer 10: To multiply, we will use distributive law as follows:
$(0.8 a - 0.5 b) (1.5 a - 3 b) = 0.8 a (1.5 a - 3 b) - 0.5 b (1.5 a - 3 b) = 1.2 a^{2} - 2.4 a b - 0.75 a b + 1.5 b^{2} = 1.2 a^{2} - 3.15 a b + 1.5 b^{2}$ Thus, the answer is $1.2 a^{2} - 3.15 a b + 1.5 b^{2}$ .

Question 11: Multiply:
(2x²y² − 5xy²) by (x² − y²)

(2x2y2−5xy2)(x2−y2)=2x2y2(x2−y2)−5xy2(x2−y2)=2x4y2−2x2y4−5x3y2+5xy4 Thus, the answer is $2 x^{4} y^{2} - 2 x^{2} y^{4} - 5 x^{3} y^{2} + 5 x y^{4}$ .

Question 15: Multiply:
(2x² − 1) by (4x³ + 5x²)

Answer 15: To multiply, we will use distributive law as follows:
$(2 x^{2} - 1) (4 x^{3} + 5 x^{2}) = 2 x^{2} (4 x^{3} + 5 x^{2}) - 1 (4 x^{3} + 5 x^{2}) = 8 x^{5} + 10 x^{4} - 4 x^{3} - 5 x^{2}$ Thus, the answer is $8 x^{5} + 10 x^{4} - 4 x^{3} - 5 x^{2}$ .

Question 16: (2xy + 3y²) (3y² − 2)

Answer 16: To multiply, we will use distributive law as follows:
$(2 x y + 3 y^{2}) (3 y^{2} - 2) = 2 x y (3 y^{2} - 2) + 3 y^{2} (3 y^{2} - 2) = 6 x y^{3} - 4 x y + 9 y^{4} - 6 y^{2} = 9 y^{4} + 6 x y^{3} - 6 y^{2} - 4 x y$ Thus, the answer is $9 y^{4} + 6 x y^{3} - 6 y^{2} - 4 x y$ .

Question 17: Find the following product and verify the result for x = − 1, y = − 2:
(3x − 5y) (x + y)

Answer 17: To multiply, we will use distributive law as follows:
$(3 x - 5 y) (x + y) = 3 x (x + y) - 5 y (x + y) = 3 x^{2} + 3 x y - 5 x y - 5 y^{2} = 3 x^{2} - 2 x y - 5 y^{2}$ $∴$ $(3 x - 5 y) (x + y) = 3 x^{2} - 2 x y - 5 y^{2}$ .

Now, we put x = $-$ 1 and y = $-$ 2 on both sides to verify the result.

LHS=(3x−5y)(x+y)={3(−1)−5(−2)}{−1+(−2)}=(−3+10)(−3)=(7)(−3)=−21 RHS=3x2−2xy−5y2=3(−1)2−2(−1)(−2)−5(−2)2=3×1−4−5×4=3−4−20=−21 Because LHS is equal to RHS, the result is verified.
Thus, the answer is $3 x^{2} - 2 x y - 5 y^{2}$ .

Question 18: Find the following product and verify the result for x = − 1, y = − 2:
(x²y − 1) (3 − 2x²y)

Question 20: Simplify:
x²(x + 2y) (x − 3y)

Question 21: Simplify:
(x² − 2y²) (x + 4y) x²y²

Question 22: Simplify:
a²b²(a + 2b)(3a + b)

Question 23: Simplify:
x²(x − y) y²(x + 2y)

Question 24: Simplify:
(x³ − 2x² + 5x − 7)(2x − 3)

Question 25: Simplify:
(5x + 3)(x − 1)(3x − 2)

Question 26: Simplify:
(5 − x)(6 − 5x)( 2 − x)

Question 27: Simplify:
(2x² + 3x − 5)(3x² − 5x + 4)

Question 28: Simplify:
(3x − 2)(2x − 3) + (5x − 3)(x + 1)

Question 29: Simplify:
(5x − 3)(x + 2) − (2x + 5)(4x − 3)

Question 30: Simplify:
(3x + 2y)(4x + 3y) − (2x − y)(7x − 3y)

Question 31: Simplify:
(x² − 3x + 2)(5x − 2) − (3x² + 4x − 5)(2x − 1)

Question 32: Simplify:
(x³ − 2x² + 3x − 4) (x −1) − (2x − 3)(x² − x + 1)