Algebraic Expressions and Identities - Exercise 6.5 | Mathematics (Maths) Class 8 PDF Download

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         ? Q u e s t i o n : 6 3
Multiply:
(5x + 3) by (7x + 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
5x+37x+2=5x7x+2+37x+2=5x×7x+5x×2+3×7x+3×2=35x2+10x+21x+6=35x2+10x+21x+6=35x2+31x+6
Thus, the answer is 35x2+31x+6.
Q u e s t i o n : 6 4
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         ?         ?         
         ?         ?   
         ?         ?         ?         ?         ?         
         ? Q u e s t i o n : 6 3
Multiply:
(5x + 3) by (7x + 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
5x+37x+2=5x7x+2+37x+2=5x×7x+5x×2+3×7x+3×2=35x2+10x+21x+6=35x2+10x+21x+6=35x2+31x+6
Thus, the answer is 35x2+31x+6.
Q u e s t i o n : 6 4
Multiply:
(2x + 8) by (x - 3)
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
2x+8x-3=2xx-3+8x-3=2x×x-2x×3+8x-8×3=2x2-6x+8x-24=2x2-6x+8x-24=2x2+2x-24
Thus, the answer is 2x2+2x-24.
Q u e s t i o n : 6 5
Multiply:
(7x + y) by (x + 5y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
7x+yx+5y=7xx+5y+yx+5y=7x2+35xy+xy+5y2=7x2+36xy+5y2
Thus, the answer is 7x2+36xy+5y2.
Q u e s t i o n : 6 6
Multiply:
(a - 1) by (0.1a
2
 + 3)
S o l u t i o n :
To multiply, we will use distributive law as follows:
a-10.1a2+3=0.1a2a-1+3a-1=0.1a3-0.1a2+3a-3
Thus, the answer is 0.1a3-0.1a2+3a-3.
Q u e s t i o n : 6 7
Multiply:
(3x
2
 + y
2
) by (2x
2
 + 3y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2+y22x2+3y2=3x22x2+3y2+y22x2+3y2=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3y4
Thus, the answer is 6x4+11x2y2+3y4.
Q u e s t i o n : 6 8
Multiply:
35x+12y by 56x+4y
S o l u t i o n :
To multiply, we will use distributive law as follows:
35x+12y56x+4y=35x56x+4y+12y56x+4y=12x2+125xy+512xy+2y2=12x2+144+2560xy+2y2=12x2+16960xy+2y2
Thus, the answer is 12x2+16960xy+2y2.
Q u e s t i o n : 6 9
Multiply:
(x
6
 - y
6
) by (x
2 
+ y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x6-y6x2+y2=x6x2+y2-y6x2+y2=x8+x6y2-y6x2+y8=x8+x6y2-y6x2-y8
Thus, the answer is x8+x6y2-y6x2-y8.
Q u e s t i o n : 7 0
Multiply:
(x
2
 + y
2
) by (3a + 2b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x2+y23a+2b=x23a+2b+y23a+2b=3ax2+2bx2+3ay2+2by2
Thus, the answer is 3ax2+2bx2+3ay2+2by2.
Q u e s t i o n : 7 1
Multiply:
[-3d + (-7f)] by (5d + f)
S o l u t i o n :
To multiply, we will use distributive law as follows:
-3d+-7f5d+f=-3d5d+f+-7f5d+f=-15d2-3df+-35df-7f2=-15d2-3df-35df-7f2=-15d2-38df-7f2
Thus, the answer is -15d2-38df-7f2.
Q u e s t i o n : 7 2
Multiply:
(0.8a - 0.5b) by (1.5a - 3b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
0.8a-0.5b1.5a-3b=0.8a1.5a-3b-0.5b1.5a-3b=1.2a2-2.4ab-0.75ab+1.5b2=1.2a2-3.15ab+1.5b2
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         ?         ?         
         ?         ?   
         ?         ?         ?         ?         ?         
         ? Q u e s t i o n : 6 3
Multiply:
(5x + 3) by (7x + 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
5x+37x+2=5x7x+2+37x+2=5x×7x+5x×2+3×7x+3×2=35x2+10x+21x+6=35x2+10x+21x+6=35x2+31x+6
Thus, the answer is 35x2+31x+6.
Q u e s t i o n : 6 4
Multiply:
(2x + 8) by (x - 3)
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
2x+8x-3=2xx-3+8x-3=2x×x-2x×3+8x-8×3=2x2-6x+8x-24=2x2-6x+8x-24=2x2+2x-24
Thus, the answer is 2x2+2x-24.
Q u e s t i o n : 6 5
Multiply:
(7x + y) by (x + 5y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
7x+yx+5y=7xx+5y+yx+5y=7x2+35xy+xy+5y2=7x2+36xy+5y2
Thus, the answer is 7x2+36xy+5y2.
Q u e s t i o n : 6 6
Multiply:
(a - 1) by (0.1a
2
 + 3)
S o l u t i o n :
To multiply, we will use distributive law as follows:
a-10.1a2+3=0.1a2a-1+3a-1=0.1a3-0.1a2+3a-3
Thus, the answer is 0.1a3-0.1a2+3a-3.
Q u e s t i o n : 6 7
Multiply:
(3x
2
 + y
2
) by (2x
2
 + 3y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2+y22x2+3y2=3x22x2+3y2+y22x2+3y2=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3y4
Thus, the answer is 6x4+11x2y2+3y4.
Q u e s t i o n : 6 8
Multiply:
35x+12y by 56x+4y
S o l u t i o n :
To multiply, we will use distributive law as follows:
35x+12y56x+4y=35x56x+4y+12y56x+4y=12x2+125xy+512xy+2y2=12x2+144+2560xy+2y2=12x2+16960xy+2y2
Thus, the answer is 12x2+16960xy+2y2.
Q u e s t i o n : 6 9
Multiply:
(x
6
 - y
6
) by (x
2 
+ y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x6-y6x2+y2=x6x2+y2-y6x2+y2=x8+x6y2-y6x2+y8=x8+x6y2-y6x2-y8
Thus, the answer is x8+x6y2-y6x2-y8.
Q u e s t i o n : 7 0
Multiply:
(x
2
 + y
2
) by (3a + 2b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x2+y23a+2b=x23a+2b+y23a+2b=3ax2+2bx2+3ay2+2by2
Thus, the answer is 3ax2+2bx2+3ay2+2by2.
Q u e s t i o n : 7 1
Multiply:
[-3d + (-7f)] by (5d + f)
S o l u t i o n :
To multiply, we will use distributive law as follows:
-3d+-7f5d+f=-3d5d+f+-7f5d+f=-15d2-3df+-35df-7f2=-15d2-3df-35df-7f2=-15d2-38df-7f2
Thus, the answer is -15d2-38df-7f2.
Q u e s t i o n : 7 2
Multiply:
(0.8a - 0.5b) by (1.5a - 3b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
0.8a-0.5b1.5a-3b=0.8a1.5a-3b-0.5b1.5a-3b=1.2a2-2.4ab-0.75ab+1.5b2=1.2a2-3.15ab+1.5b2
Thus, the answer is 1.2a2-3.15ab+1.5b2.
Q u e s t i o n : 7 3
Multiply:
(2x
2
y
2
 - 5xy
2
) by (x
2
 - y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2x2y2-5xy2x2-y2=2x2y2x2-y2-5xy2x2-y2=2x4y2-2x2y4-5x3y2+5xy4
Thus, the answer is 2x4y2-2x2y4-5x3y2+5xy4.
Q u e s t i o n : 7 4
Multiply:
x7+x22by25+9x4
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
x7+x2225+9x4=x725+9x4+x2225+9x4=2x35+9x228+x25+9x38=2x35+45+28140x2+ 9x38=2x35+73x2140 +9x28
Thus, the answer is  2x35+73x2140+9x38
Q u e s t i o n : 7 5
Multiply:
-a7+a29byb2-b23
S o l u t i o n :
To multiply, we will use distributive law as follows:
-a7+a29b2-b23=-a7b2-b23+a29b2-b23=-ab14+ab221+a2b18-a2b227=-ab14+ab221+a2b18-a2b227
Thus, the answer is -ab14+ab221+a2b18-a2b227.
Q u e s t i o n : 7 6
Multiply:
(3x
2
y - 5xy
2
) by 15x2 + 13y2
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2y-5xy215x2+13y2=15x23x2y-5xy2+13y23x2y-5xy2=35x4y-x3y2+x2y3-53xy4
Thus, the answer is 35x4y-x3y2+x2y3-53xy4.
Q u e s t i o n : 7 7
Multiply:
(2x
2
 - 1) by (4x
3
 + 5x
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2x2-14x3+5x2=2x24x3+5x2-14x3+5x2=8x5+10x4-4x3-5x2
Thus, the answer is 8x5+10x4-4x3-5x2.
Q u e s t i o n : 7 8
(2xy + 3y
2
) (3y
2
 - 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2xy+3y23y2-2=2xy3y2-2+3y23y2-2=6xy3-4xy+9y4-6y2=9y4+6xy3-6y2-4xy
Thus, the answer is 9y4+6xy3-6y2-4xy.
Q u e s t i o n : 7 9
Find the following product and verify the result for x = - 1, y = - 2:
(3x - 5y) (x + y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x-5yx+y=3xx+y-5yx+y=3x2+3xy-5xy-5y2=3x2-2xy-5y2
? 3x-5yx+y=3x2-2xy-5y2.
Now, we put x = -1 and y = -2 on both sides to verify the result.
LHS=3x-5yx+y=3-1-5-2-1+-2=-3+10-3=7-3=-21
RHS=3x2-2xy-5y2=3-12-2-1-2-5-22=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-5y2.
Q u e s t i o n : 8 0
Find the following product and verify the result for x = - 1, y = - 2:
(x
2
y - 1) (3 - 2x
2
y)
S o l u t i o n :
To multiply, we will  use distributive law as follows:
Page 4


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         ?         ?         
         ?         ?   
         ?         ?         ?         ?         ?         
         ? Q u e s t i o n : 6 3
Multiply:
(5x + 3) by (7x + 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
5x+37x+2=5x7x+2+37x+2=5x×7x+5x×2+3×7x+3×2=35x2+10x+21x+6=35x2+10x+21x+6=35x2+31x+6
Thus, the answer is 35x2+31x+6.
Q u e s t i o n : 6 4
Multiply:
(2x + 8) by (x - 3)
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
2x+8x-3=2xx-3+8x-3=2x×x-2x×3+8x-8×3=2x2-6x+8x-24=2x2-6x+8x-24=2x2+2x-24
Thus, the answer is 2x2+2x-24.
Q u e s t i o n : 6 5
Multiply:
(7x + y) by (x + 5y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
7x+yx+5y=7xx+5y+yx+5y=7x2+35xy+xy+5y2=7x2+36xy+5y2
Thus, the answer is 7x2+36xy+5y2.
Q u e s t i o n : 6 6
Multiply:
(a - 1) by (0.1a
2
 + 3)
S o l u t i o n :
To multiply, we will use distributive law as follows:
a-10.1a2+3=0.1a2a-1+3a-1=0.1a3-0.1a2+3a-3
Thus, the answer is 0.1a3-0.1a2+3a-3.
Q u e s t i o n : 6 7
Multiply:
(3x
2
 + y
2
) by (2x
2
 + 3y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2+y22x2+3y2=3x22x2+3y2+y22x2+3y2=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3y4
Thus, the answer is 6x4+11x2y2+3y4.
Q u e s t i o n : 6 8
Multiply:
35x+12y by 56x+4y
S o l u t i o n :
To multiply, we will use distributive law as follows:
35x+12y56x+4y=35x56x+4y+12y56x+4y=12x2+125xy+512xy+2y2=12x2+144+2560xy+2y2=12x2+16960xy+2y2
Thus, the answer is 12x2+16960xy+2y2.
Q u e s t i o n : 6 9
Multiply:
(x
6
 - y
6
) by (x
2 
+ y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x6-y6x2+y2=x6x2+y2-y6x2+y2=x8+x6y2-y6x2+y8=x8+x6y2-y6x2-y8
Thus, the answer is x8+x6y2-y6x2-y8.
Q u e s t i o n : 7 0
Multiply:
(x
2
 + y
2
) by (3a + 2b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x2+y23a+2b=x23a+2b+y23a+2b=3ax2+2bx2+3ay2+2by2
Thus, the answer is 3ax2+2bx2+3ay2+2by2.
Q u e s t i o n : 7 1
Multiply:
[-3d + (-7f)] by (5d + f)
S o l u t i o n :
To multiply, we will use distributive law as follows:
-3d+-7f5d+f=-3d5d+f+-7f5d+f=-15d2-3df+-35df-7f2=-15d2-3df-35df-7f2=-15d2-38df-7f2
Thus, the answer is -15d2-38df-7f2.
Q u e s t i o n : 7 2
Multiply:
(0.8a - 0.5b) by (1.5a - 3b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
0.8a-0.5b1.5a-3b=0.8a1.5a-3b-0.5b1.5a-3b=1.2a2-2.4ab-0.75ab+1.5b2=1.2a2-3.15ab+1.5b2
Thus, the answer is 1.2a2-3.15ab+1.5b2.
Q u e s t i o n : 7 3
Multiply:
(2x
2
y
2
 - 5xy
2
) by (x
2
 - y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2x2y2-5xy2x2-y2=2x2y2x2-y2-5xy2x2-y2=2x4y2-2x2y4-5x3y2+5xy4
Thus, the answer is 2x4y2-2x2y4-5x3y2+5xy4.
Q u e s t i o n : 7 4
Multiply:
x7+x22by25+9x4
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
x7+x2225+9x4=x725+9x4+x2225+9x4=2x35+9x228+x25+9x38=2x35+45+28140x2+ 9x38=2x35+73x2140 +9x28
Thus, the answer is  2x35+73x2140+9x38
Q u e s t i o n : 7 5
Multiply:
-a7+a29byb2-b23
S o l u t i o n :
To multiply, we will use distributive law as follows:
-a7+a29b2-b23=-a7b2-b23+a29b2-b23=-ab14+ab221+a2b18-a2b227=-ab14+ab221+a2b18-a2b227
Thus, the answer is -ab14+ab221+a2b18-a2b227.
Q u e s t i o n : 7 6
Multiply:
(3x
2
y - 5xy
2
) by 15x2 + 13y2
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2y-5xy215x2+13y2=15x23x2y-5xy2+13y23x2y-5xy2=35x4y-x3y2+x2y3-53xy4
Thus, the answer is 35x4y-x3y2+x2y3-53xy4.
Q u e s t i o n : 7 7
Multiply:
(2x
2
 - 1) by (4x
3
 + 5x
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2x2-14x3+5x2=2x24x3+5x2-14x3+5x2=8x5+10x4-4x3-5x2
Thus, the answer is 8x5+10x4-4x3-5x2.
Q u e s t i o n : 7 8
(2xy + 3y
2
) (3y
2
 - 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2xy+3y23y2-2=2xy3y2-2+3y23y2-2=6xy3-4xy+9y4-6y2=9y4+6xy3-6y2-4xy
Thus, the answer is 9y4+6xy3-6y2-4xy.
Q u e s t i o n : 7 9
Find the following product and verify the result for x = - 1, y = - 2:
(3x - 5y) (x + y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x-5yx+y=3xx+y-5yx+y=3x2+3xy-5xy-5y2=3x2-2xy-5y2
? 3x-5yx+y=3x2-2xy-5y2.
Now, we put x = -1 and y = -2 on both sides to verify the result.
LHS=3x-5yx+y=3-1-5-2-1+-2=-3+10-3=7-3=-21
RHS=3x2-2xy-5y2=3-12-2-1-2-5-22=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-5y2.
Q u e s t i o n : 8 0
Find the following product and verify the result for x = - 1, y = - 2:
(x
2
y - 1) (3 - 2x
2
y)
S o l u t i o n :
To multiply, we will  use distributive law as follows:
x2y-13-2x2y=x2y3-2x2y-1×3-2x2y=3x2y-2x4y2-3+2x2y=5x2y-2x4y2-3
? x2y-13-2x2y=5x2y-2x4y2-3
Now, we put x = -1 and y = -2 on both sides to verify the result.
LHS = x2y-13-2x2y=-12-2-13-2-12-2=1×-2-13-2×1×-2=-2-13+4=-3×7=-21
RHS=5x2y-2x4y2-3=5-12-2-2-14-22-3=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 5x2y-2x4y2-3.
Q u e s t i o n : 8 1
Find the following product and verify the result for x = - 1, y = - 2:
13x-y2513x+y25
S o l u t i o n :
To multiply, we will use distributive law as follows:
13x-y2513x+y25=13x13x+y25-y2513x+y25=19x2+xy215-xy215+y425=19x2+xy215-xy215-y425=19x2-y425
? 13x-y2513x+y25=19x2-y425
Now, we will put x = -1 and y = -2 on both the sides to verify the result.
LHS = 13x-y2513x+y25=13-1--22513-1+-225=-13-45-13+45=-1715715=-119225
RHS=19x2-y425=19-12--2425=19×1-1625=19-1625=-119225
Because LHS is equal to RHS, the result is verified.
Thus, the answer is 19x2-y425.
Q u e s t i o n : 8 2
Simplify:
x
2
(x + 2y) (x - 3y)
S o l u t i o n :
To simplify, we will proceed as follows:
x2x+2yx-3y=x2x+2yx-3y=x3+2x2yx-3y=x3x-3y+2x2yx-3y=x4-3x3y+2x3y-6x2y2=x4-x3y-6x2y2
Thus, the answer is x4-x3y-6x2y2.
Q u e s t i o n : 8 3
Simplify:
(x
2
 - 2y
2
) (x + 4y) x
2
y
2
S o l u t i o n :
To simplify, we will proceed as follows:
x2-2y2x+4yx2y2=x2x+4y-2y2x+4yx2y2=x3+4x2y-2xy2-8y3x2y2=x5y2+4x4y3-2x3y4-8x2y5
Thus, the answer is x5y2+4x4y3-2x3y4-8x2y5.
Q u e s t i o n : 8 4
Simplify:
a
2
b
2
(a + 2b)(3a + b)
S o l u t i o n :
To simplify, we will proceed as follows:
a2b2a+2b3a+b=a2b2a+2b3a+b=a3b2+2a2b33a+b=3aa3b2+2a2b3+ba3b2+2a2b3=3a4b2+6a3b3+a3b3+2a2b4=3a4b2+7a3b3+2a2b4
Thus, the answer is 3a4b2+7a3b3+2a2b4.
Q u e s t i o n : 8 5
Simplify:
x
2
(x - y) y
2
(x + 2y)
S o l u t i o n :
To simplify, we will proceed as follows:
x2x-yy2x+2y=x2x-yy2x+2y=x3-x2yxy2+2y3=x3xy2+2y3-x2yxy2+2y3=x4y2+2x3y3-x3y3+2x2y4=x4y2+2x3y3-x3y3-2x2y4=x4y2+x3y3-2x2y4
Thus, the answer is x4y2+x3y3-2x2y4.
Q u e s t i o n : 8 6
Simplify:
(x
3
 - 2x
2
 + 5x - 7)(2x - 3)
S o l u t i o n :
To simplify, we will proceed as follows:
x3-2x2+5x-72x-3=2xx3-2x2+5x-7-3x3-2x2+5x-7=2x4-4x3+10x2-14x-3x3+6x2-15x+21
=2x4-4x3-3x3+10x2+6x2-14x-15x+21     Rearranging
=2x4-7x3+16x2-29x+21                              Combining like terms
Thus, the answer is 2x4-7x3+16x2-29x+21.
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         ? Q u e s t i o n : 6 3
Multiply:
(5x + 3) by (7x + 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
5x+37x+2=5x7x+2+37x+2=5x×7x+5x×2+3×7x+3×2=35x2+10x+21x+6=35x2+10x+21x+6=35x2+31x+6
Thus, the answer is 35x2+31x+6.
Q u e s t i o n : 6 4
Multiply:
(2x + 8) by (x - 3)
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
2x+8x-3=2xx-3+8x-3=2x×x-2x×3+8x-8×3=2x2-6x+8x-24=2x2-6x+8x-24=2x2+2x-24
Thus, the answer is 2x2+2x-24.
Q u e s t i o n : 6 5
Multiply:
(7x + y) by (x + 5y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
7x+yx+5y=7xx+5y+yx+5y=7x2+35xy+xy+5y2=7x2+36xy+5y2
Thus, the answer is 7x2+36xy+5y2.
Q u e s t i o n : 6 6
Multiply:
(a - 1) by (0.1a
2
 + 3)
S o l u t i o n :
To multiply, we will use distributive law as follows:
a-10.1a2+3=0.1a2a-1+3a-1=0.1a3-0.1a2+3a-3
Thus, the answer is 0.1a3-0.1a2+3a-3.
Q u e s t i o n : 6 7
Multiply:
(3x
2
 + y
2
) by (2x
2
 + 3y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2+y22x2+3y2=3x22x2+3y2+y22x2+3y2=6x4+9x2y2+2x2y2+3y4=6x4+11x2y2+3y4
Thus, the answer is 6x4+11x2y2+3y4.
Q u e s t i o n : 6 8
Multiply:
35x+12y by 56x+4y
S o l u t i o n :
To multiply, we will use distributive law as follows:
35x+12y56x+4y=35x56x+4y+12y56x+4y=12x2+125xy+512xy+2y2=12x2+144+2560xy+2y2=12x2+16960xy+2y2
Thus, the answer is 12x2+16960xy+2y2.
Q u e s t i o n : 6 9
Multiply:
(x
6
 - y
6
) by (x
2 
+ y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x6-y6x2+y2=x6x2+y2-y6x2+y2=x8+x6y2-y6x2+y8=x8+x6y2-y6x2-y8
Thus, the answer is x8+x6y2-y6x2-y8.
Q u e s t i o n : 7 0
Multiply:
(x
2
 + y
2
) by (3a + 2b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
x2+y23a+2b=x23a+2b+y23a+2b=3ax2+2bx2+3ay2+2by2
Thus, the answer is 3ax2+2bx2+3ay2+2by2.
Q u e s t i o n : 7 1
Multiply:
[-3d + (-7f)] by (5d + f)
S o l u t i o n :
To multiply, we will use distributive law as follows:
-3d+-7f5d+f=-3d5d+f+-7f5d+f=-15d2-3df+-35df-7f2=-15d2-3df-35df-7f2=-15d2-38df-7f2
Thus, the answer is -15d2-38df-7f2.
Q u e s t i o n : 7 2
Multiply:
(0.8a - 0.5b) by (1.5a - 3b)
S o l u t i o n :
To multiply, we will use distributive law as follows:
0.8a-0.5b1.5a-3b=0.8a1.5a-3b-0.5b1.5a-3b=1.2a2-2.4ab-0.75ab+1.5b2=1.2a2-3.15ab+1.5b2
Thus, the answer is 1.2a2-3.15ab+1.5b2.
Q u e s t i o n : 7 3
Multiply:
(2x
2
y
2
 - 5xy
2
) by (x
2
 - y
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2x2y2-5xy2x2-y2=2x2y2x2-y2-5xy2x2-y2=2x4y2-2x2y4-5x3y2+5xy4
Thus, the answer is 2x4y2-2x2y4-5x3y2+5xy4.
Q u e s t i o n : 7 4
Multiply:
x7+x22by25+9x4
S o l u t i o n :
To multiply the expressions, we will use the distributive law in the following way:
x7+x2225+9x4=x725+9x4+x2225+9x4=2x35+9x228+x25+9x38=2x35+45+28140x2+ 9x38=2x35+73x2140 +9x28
Thus, the answer is  2x35+73x2140+9x38
Q u e s t i o n : 7 5
Multiply:
-a7+a29byb2-b23
S o l u t i o n :
To multiply, we will use distributive law as follows:
-a7+a29b2-b23=-a7b2-b23+a29b2-b23=-ab14+ab221+a2b18-a2b227=-ab14+ab221+a2b18-a2b227
Thus, the answer is -ab14+ab221+a2b18-a2b227.
Q u e s t i o n : 7 6
Multiply:
(3x
2
y - 5xy
2
) by 15x2 + 13y2
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x2y-5xy215x2+13y2=15x23x2y-5xy2+13y23x2y-5xy2=35x4y-x3y2+x2y3-53xy4
Thus, the answer is 35x4y-x3y2+x2y3-53xy4.
Q u e s t i o n : 7 7
Multiply:
(2x
2
 - 1) by (4x
3
 + 5x
2
)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2x2-14x3+5x2=2x24x3+5x2-14x3+5x2=8x5+10x4-4x3-5x2
Thus, the answer is 8x5+10x4-4x3-5x2.
Q u e s t i o n : 7 8
(2xy + 3y
2
) (3y
2
 - 2)
S o l u t i o n :
To multiply, we will use distributive law as follows:
2xy+3y23y2-2=2xy3y2-2+3y23y2-2=6xy3-4xy+9y4-6y2=9y4+6xy3-6y2-4xy
Thus, the answer is 9y4+6xy3-6y2-4xy.
Q u e s t i o n : 7 9
Find the following product and verify the result for x = - 1, y = - 2:
(3x - 5y) (x + y)
S o l u t i o n :
To multiply, we will use distributive law as follows:
3x-5yx+y=3xx+y-5yx+y=3x2+3xy-5xy-5y2=3x2-2xy-5y2
? 3x-5yx+y=3x2-2xy-5y2.
Now, we put x = -1 and y = -2 on both sides to verify the result.
LHS=3x-5yx+y=3-1-5-2-1+-2=-3+10-3=7-3=-21
RHS=3x2-2xy-5y2=3-12-2-1-2-5-22=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-5y2.
Q u e s t i o n : 8 0
Find the following product and verify the result for x = - 1, y = - 2:
(x
2
y - 1) (3 - 2x
2
y)
S o l u t i o n :
To multiply, we will  use distributive law as follows:
x2y-13-2x2y=x2y3-2x2y-1×3-2x2y=3x2y-2x4y2-3+2x2y=5x2y-2x4y2-3
? x2y-13-2x2y=5x2y-2x4y2-3
Now, we put x = -1 and y = -2 on both sides to verify the result.
LHS = x2y-13-2x2y=-12-2-13-2-12-2=1×-2-13-2×1×-2=-2-13+4=-3×7=-21
RHS=5x2y-2x4y2-3=5-12-2-2-14-22-3=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 5x2y-2x4y2-3.
Q u e s t i o n : 8 1
Find the following product and verify the result for x = - 1, y = - 2:
13x-y2513x+y25
S o l u t i o n :
To multiply, we will use distributive law as follows:
13x-y2513x+y25=13x13x+y25-y2513x+y25=19x2+xy215-xy215+y425=19x2+xy215-xy215-y425=19x2-y425
? 13x-y2513x+y25=19x2-y425
Now, we will put x = -1 and y = -2 on both the sides to verify the result.
LHS = 13x-y2513x+y25=13-1--22513-1+-225=-13-45-13+45=-1715715=-119225
RHS=19x2-y425=19-12--2425=19×1-1625=19-1625=-119225
Because LHS is equal to RHS, the result is verified.
Thus, the answer is 19x2-y425.
Q u e s t i o n : 8 2
Simplify:
x
2
(x + 2y) (x - 3y)
S o l u t i o n :
To simplify, we will proceed as follows:
x2x+2yx-3y=x2x+2yx-3y=x3+2x2yx-3y=x3x-3y+2x2yx-3y=x4-3x3y+2x3y-6x2y2=x4-x3y-6x2y2
Thus, the answer is x4-x3y-6x2y2.
Q u e s t i o n : 8 3
Simplify:
(x
2
 - 2y
2
) (x + 4y) x
2
y
2
S o l u t i o n :
To simplify, we will proceed as follows:
x2-2y2x+4yx2y2=x2x+4y-2y2x+4yx2y2=x3+4x2y-2xy2-8y3x2y2=x5y2+4x4y3-2x3y4-8x2y5
Thus, the answer is x5y2+4x4y3-2x3y4-8x2y5.
Q u e s t i o n : 8 4
Simplify:
a
2
b
2
(a + 2b)(3a + b)
S o l u t i o n :
To simplify, we will proceed as follows:
a2b2a+2b3a+b=a2b2a+2b3a+b=a3b2+2a2b33a+b=3aa3b2+2a2b3+ba3b2+2a2b3=3a4b2+6a3b3+a3b3+2a2b4=3a4b2+7a3b3+2a2b4
Thus, the answer is 3a4b2+7a3b3+2a2b4.
Q u e s t i o n : 8 5
Simplify:
x
2
(x - y) y
2
(x + 2y)
S o l u t i o n :
To simplify, we will proceed as follows:
x2x-yy2x+2y=x2x-yy2x+2y=x3-x2yxy2+2y3=x3xy2+2y3-x2yxy2+2y3=x4y2+2x3y3-x3y3+2x2y4=x4y2+2x3y3-x3y3-2x2y4=x4y2+x3y3-2x2y4
Thus, the answer is x4y2+x3y3-2x2y4.
Q u e s t i o n : 8 6
Simplify:
(x
3
 - 2x
2
 + 5x - 7)(2x - 3)
S o l u t i o n :
To simplify, we will proceed as follows:
x3-2x2+5x-72x-3=2xx3-2x2+5x-7-3x3-2x2+5x-7=2x4-4x3+10x2-14x-3x3+6x2-15x+21
=2x4-4x3-3x3+10x2+6x2-14x-15x+21     Rearranging
=2x4-7x3+16x2-29x+21                              Combining like terms
Thus, the answer is 2x4-7x3+16x2-29x+21.
Q u e s t i o n : 8 7
Simplify:
(5x + 3)(x - 1)(3x - 2)
S o l u t i o n :
To simplify, we will proceed as follows:
5x+3x-13x-2=5x+3x-13x-2
=5xx-1+3x-13x-2              Distributive law
=5x2-5x+3x-33x-2=5x2-2x-33x-2=3x5x2-2x-3-25x2-2x-3=15x3-6x2-9x-10x2-4x-6=15x3-6x2-9x-10x2+4x+6
=15x3-6x2-10x2-9x+4x+6              Rearranging
=15x3-16x2-5x+6                              Combining like terms
Thus, the answer is 15x3-16x2-5x+6.
Q u e s t i o n : 8 8
Simplify:
(5 - x)(6 - 5x)( 2 - x)
S o l u t i o n :
To simplify, we will proceed as follows:
5-x6-5x2-x=5-x6-5x2-x
=56-5x-x6-5x2-x                Distributive law
=30-25x-6x+5x22-x=30-31x+5x22-x=230-31x+5x2-x30-31x+5x2=60-62x+10x2-30x+31x2-5x3
=60-62x-30x+10x2+31x2-5x3              Rearranging
=60-92x+41x2-5x3                                Combining like terms
Thus, the answer is 60-92x+41x2-5x3.
Q u e s t i o n : 8 9
Simplify:
(2x
2
 + 3x - 5)(3x
2
 - 5x + 4)
S o l u t i o n :
To simplify, we will proceed as follows:
2x2+3x-53x2-5x+4
=2x23x2-5x+4+3x3x2-5x+4-53x2-5x+4           Distributive law
=6x4-10x3+8x2+9x3-15x2+12x-15x2+25x-20
=6x4-10x3+9x3+8x2-15x2-15x2+12x+25x-20              Rearranging
=6x4-x3-22x2+36x-20                                                     Combining like terms
Thus, the answer is 6x4-x3-22x2+36x-20.
Q u e s t i o n : 9 0
Simplify:
(3x - 2)(2x - 3) + (5x - 3)(x + 1)
S o l u t i o n :
To simplify, we will proceed as follows:
3x-22x-3+5x-3x+1=3x-22x-3+5x-3x+1
=3x2x-3-22x-3+5xx+1-3x+1           Distributive law
=6x2-9x-4x+6+5x2+5x-3x-3
=6x2+5x2-9x-4x+5x-3x-3+6                                  Rearranging
=11x2-11x+3                                                                 Combining like terms
Thus, the answer is 11x2-11x+3.
Q u e s t i o n : 9 1
Simplify:
(5x - 3)(x + 2) - (2x + 5)(4x - 3)
S o l u t i o n :
To simplify, we will proceed as follows:
5x-3x+2-2x+54x-3=5x-3x+2-2x+54x-3
=5xx+2-3x+2-2x4x-3+54x-3            Distributive law
=5x2+10x-3x-6-8x2+6x-20x+15
=5x2-8x2+10x-3x+6x-20x-6+15                              Rearranging
=5x2-8x2+10x-3x+6x-20x-6+15=-3x2-7x+9                              Combining like terms
Hence, the answer is -3x2-7x+9.
Q u e s t i o n : 9 2
Simplify:
(3x + 2y)(4x + 3y) - (2x - y)(7x - 3y)
S o l u t i o n :
To simplify, we will proceed as follows:
3x+2y4x+3y-2x-y7x-3y=3x+2y4x+3y-2x-y7x-3y
=3x4x+3y+2y4x+3y-2x7x-3y-y7x-3y            Distributive law
=12x2+9xy+8xy+6y2-14x2-6xy-7xy+3y2=12x2+9xy+8xy+6y2-14x2+6xy+7xy-3y2
=12x2-14x2+9xy+8xy+6xy+7xy+6y2-3y2                              Rearranging
=-2x2+30xy+3y2                                                                       Combining like terms
Thus, the answer is -2x2+30xy+3y2.
Q u e s t i o n : 9 3
Simplify:
(x
2
 - 3x + 2)(5x - 2) - (3x
2
 + 4x - 5)(2x - 1)
S o l u t i o n :