Problem 1 :
Add (3x2 + 7x - 2) and (x2 - 4x + 5).
Problem 2 :
Add (7y3 - 5y2 + 1) and (2y2 - y - 7).
Problem 3 :
Add (2p - q), (p + 3q) and (p - q).
Problem 4 :
Subtract 3ab from 7ab.
Problem 5 :
Subtract (2x + 3y2) from (5x - 2y2).
Problem 6 :
Add (a3 + 3a2 - 5a - 2) and (a3 - 6a2 + 9a - 8).
Problem 7 :
Add (2x3 + 5x2 - 2x + 7) and (x3 + 4x2 - x + 6).
Problem 8 :
Subtract (3x3 - 5y2 + 1) from (4x3 - 2y2 + 1).
Problem 9 :
Simplify :
(3x3 - 2x2 - x + 4) + (2x3 + 7x2 - 3x - 3)
Problem 10 :
Simplify :
(y3 + 2y2 - 3y + 1) + 3(2y5 - 7y4 + 2y - 5)
Problem 11 :
Simplify :
-2(p5 + 2p4 - p2 + 5) + 3(4p6 - 4p4 - 1)
Problem 12 :
Simplify :
5(5x6 + 2x3 - 6x2 - 2) + 6(-3x6 + 2x5 + 2x + 1).
Problem 13 :
Simplify :
-2(2x4 - 2x3 - x2 + 5) + 3(2x4 - 2x2 - 3).
Problem 14 :
Simplify :
5(x4 - x3 + 5) + 2(x4 - 5x2 - 7).
Problem 15 :
Simplify :
3(6x4 - 2x3 - 3) + 2(2x4 - x2 - 8).
Problem 16 :
Simplify :
(6x7 - 2x6 - 3x3 + 2x2) + 2(2x4 + 5x7 + 3x6 + x3 + x2)
Problem 17 :
Simplify :
(x7 - 3x6 - 2x3 + x2) + 5(3x4 + 15x7 + 4x6 + 2x3 + 6x2)
Problem 18 :
Subtract (2x3 + 5x2 - 2x - 11) from (3x3 - 2x2 - 5x - 6).
Problem 19 :
Subtract (x3 + 4x2 - 12x - 5) from (5x3 + 3x2 + 2x - 10).
Problem 20 :
Subtract (2x3 - 5x2 + 7x - 3) from 3(5x3 - 2x + 8).
Problem 21 :
Subtract 2(y2 + 3y - 1) from 5(2y3 - 2y2 - 7y - 4).
Problem 22 :
Simplify :
(x2 + y2) - (x + y)(x - y)
Problem 23 :
Simplify :
(p - q)3 - (p + q)3
1. Answer :
= (3x2 + 7x - 2) + (x2 - 4x + 5)
= 3x2 + 7x - 2 + x2 - 4x + 5
Group the like terms together.
= (3x2 + x2) + (7x - 4x) + (-2 + 5)
Combine like terms.
= 4x2 + 3x + 3
2. Answer :
= (7y3 - 5y2 + 1) + (2y2 - y - 7)
= 7y3 - 5y2 + 1 + 2y2 - y - 7
Group the like terms together.
= 7y3 + (-5y2 + 2y2) - y + (1 - 7)
Combine the like terms.
= 7y3 + (-3y2) - y + (-6)
= 7y3 - 3y2 - y - 6
3. Answer :
= (2p - q) + (p + 3q) + (p - q)
= 2p - q + p + 3q + p - q
Group the like terms together.
= (2p + p + p) + (-q + 3q - q)
Combine the like terms.
= 4p + q
4. Answer :
= 7ab - 3ab
Both the terms in the above expression are like terms. So, we can combine them.
= 4ab
5. Answer :
= (5x - 2y2) - (2x + 3y2)
Distribute the negative sign.
= 5x - 2y2 - 2x - 3y2
Group the like terms together.
= (5x - 2x) + (-2y2 - 3y2)
Combine the like terms.
= 3x + (-5y2)
= 3x - 5y2
6. Answer :
= (a3 + 3a2 - 5a - 2) + (a3 - 6a2 + 9a - 8)
= a3 + 3a2 - 5a - 2 + a3 - 6a2 + 9a - 8
Group the like terms together.
= (a3 + a3) + (3a2 - 6a2) + (-5a + 9a) + (-2 - 8)
Combine the like terms.
= 2a3 + (-3a2) + 4a + (-10)
= 2a3 - 3a2 + 4a - 10
7. Answer :
= (2x3 + 5x2 - 2x + 7) + (x3 + 4x2 - x + 6)
= 2x3 + 5x2 - 2x + 7 + x3 + 4x2 - x + 6
Group like terms together.
= (2x3 + x3) + (5x2 + 4x2) + (-2x - x) + (7 + 6)
Combine like terms.
= 3x3 + 9x2 - 3x + 13
8. Answer :
= (4x3 - 2y2 + 1) - (3x3 - 5y2 + 1)
Distribute the negative sign.
= 4x3 - 2y2 + 1 - 3x3 + 5y2 - 1
Group the like terms together.
= (4x3 - 3x3) + (-2y2 + 5y2) + (1 - 1)
Combine the like terms.
= x3 + 3y2 + 0
= x3 + 3y2
9. Answer :
= (3x3 - 2x2 - x + 4) + (2x3 + 7x2 - 3x - 3)
= 3x3 - 2x2 - x + 4 + 2x3 + 7x2 - 3 x - 3
Group like terms together.
= (3x3 + 2x3) + (-2x2 + 7x2) + (-x - 3x) + (4 - 3)
Combine like terms.
= 5x3 + 5x2 - 4x + 1
10. Answer :
= (y3 + 2y2 - 3y + 1) + 3(2y5 - 7y4 + 2y - 5)
Using Distributive Property,
= (y3 + 2y2 - 3y + 1) + 3(2y5 - 7y4 + 2y - 5)
= y3 + 2y2 - 3y + 1 + 3(2y5) + 3(-7y4) + 3(2y) + 3(-5)
= y3 + 2y2 - 3y + 1 + 6y5 - 21y4 + 6y - 15
= 6y5 - 21y4 + y3 + 2y2 + (-3y + 6y) + (1 - 15)
= 6y5 - 21y4 + y3 + 2y2 + 3y + (-14)
= 6y5 - 21y4 + y3 + 2y2 + 3y - 14
11. Answer :
= -2(p5 + 2p4 - p2 + 5) + 3(4p6 - 4p4 - 1)
= -2(p5) - 2(2p4) - 2(-p2) - 2(+5) + 3(4p6) + 3(-4p4) + 3(-1)
= -2p5 - 4p4 + 2p2 - 10 + 12p6 - 12p4 - 3
= 12p6 - 2p5 + (-4p4 - 12p4) + 2p2 + (-10 - 3)
= 12p6 - 2p5 + (-16p4) + 2p2 + (-13)
= 12p6 - 2p5 - 16p4 + 2p2 - 13
12. Answer :
= 5(5x6 + 2x3 - 6x2 - 2) + 6(-3x6 + 2x5 + 2x + 1)
Distributive Property.
= 25x6 + 10x3 - 30x2 - 10 - 18x6 + 12x5 + 12x + 6
Group like terms together.
= (25x6 - 18x6) + 12x5 + 10x3 - 30x2 + 12x + (-10 + 6)
Combine like terms.
= 7x6 + 12x5 + 10x3 - 30x2 + 12x - 4
13. Answer :
= -2(2x4 - 2x3 - x2 + 5) + 3(2x4 - 2x2 - 3)
Distributive Property.
= -4x4 + 4x3 + 2x2 -10 + 6x4 - 6x2 - 9
Group like terms together.
= (-4x4 + 6x4) + 4x3 + (2x2 - 6x2) + (-10 - 9)
Combine like terms.
= 2x4 + 4x3 - 4x2 - 19
14. Answer :
= 5(x4 - x3 + 5) + 2(x4 - 5x2 - 7)
Distributive Property.
= 5x4 - 5x3 + 25 + 2x4 - 10x2 - 14
Group like terms together.
= (5x4 + 2x4) - 5x3 - 10x2 + (25 - 14)
Combine like terms.
= 7x4 - 5x3 - 10x2 + 11
15. Answer :
= 3(6x4 - 2x3 - 3) + 2(2x4 - x2 - 8)
Distributive Property.
= 18x4 - 6x3 - 9 + 4x4 - 2x2 - 16
Group like terms together.
= (18x4 + 4x4) - 6x3 - 2x2 + (-9 - 16)
Combine like terms.
= 22x4 - 6x3 - 2x2 - 25
16. Answer :
= (6x7 - 2x6 - 3x3 + 2x2) + 2(2x4 + 5x7 + 3x6 + x3 + x2)
Distributive Property.
= 6x7 - 2x6 - 3x3 + 2x2 + 4x4 + 10x7 + 6x6 + 2x3 + 2x2
Group like terms together.
= (6x7 + 10x7) + (-2x6 + 6x6) + 4x4 + (-3x3 + 2x3) + (2x2 + 2x2)
Combine like terms.
= 16x7 + 4x6 + 4x4 - x3 + 4x2
17. Answer :
= (x7 - 3x6 - 2x3 + x2) + 5(3x4 + 15x7 + 4x6 + 2x3 + 6x2)
Distributive Property.
= x7 - 3x6 - 2x3 + x2 + 15x4 + 75x7 + 20x6 + 10x3 + 30x2
Group like terms together.
= (x7 + 75x7) + (-3x6 + 20x6) + (-2x3 + 10x3) + 15x4 + (x2 + 30x2)
Combine like terms.
= 76x7 + 17x6 + 8x3 + 15x4 + 31x2
18. Answer :
= (3x3 - 2x2 - 5x - 6 ) - (2x3 + 5x2 - 2x - 11)
Distributive Property.
= 3x3 - 2x2 - 5x - 6 -2x3 - 5x2 + 2x + 11
Group like terms together.
= (3x3 - 2x3) + (-2x2 - 5x2) + (-5x + 2x) + (-6 + 11)
Combine like terms.
= x3 - 7x2 - 3x + 5
19. Answer :
= (5x3 + 3x2 + 2x - 10) - (x3 + 4x2 - 12x - 5)
Distributive Property.
= 5x3 + 3x2 + 2x - 10 - x3 - 4x2 + 12x + 5
Group like terms together.
= (5x3 - x3) + (3x2 - 4x2) + (2x + 12x) + (-10 + 5)
Combine like terms.
= 4x3 - x2 + 14x - 5
20. Answer :
= 3(5x3 - 2x + 8) - (2x3 - 5x2 + 7x - 3)
Using Distributive Property,
= 3(5x3) + 3(-2x) + 3(+8) - 2x3 + 5x2 - 7x + 3
= 15x3 - 6x + 24 - 2x3 + 5x2 - 7x + 3
= (15x3 - 2x3) + 5x2 + (-6x - 7x) + (24 + 3)
= 13x3 + 5x2 + (-13x) + 27
= 13x3 + 5x2 - 13x + 27
21. Answer :
= 5(2y3 - 2y2 - 7y - 4) - 2(y2 + 3y - 1)
Using Distributive Property,
= 5(2y3 - 2y2 - 7y - 4) - 2(y2 + 3y - 1)
= 5(2y3) + 5(-2y2) + 5(-7y) + 5(-4) - 2(y2) - 2(+3y) - 2(-1)
= 10y3 - 10y2 - 35y - 20 - 2y2 - 6y + 2
= 10y3 + (-10y2 - 2y2) + (-35y - 6y) + (-20 + 2)
= 10y3 + (-12y2) + (-41y) + (-18)
= 10y3 - 12y2 - 41y - 18
22. Answer :
= (x2 + y2) - (x + y)(x - y)
= x2 + y2 - [(x + y)(x - y)]
= x2 + y2 - [x2 - xy + xy - y2]
= x2 + y2 - [x2 - y2]
= x2 + y2 - x2 + y2
= 2y2
23. Answer :
The following algebraic identities can be used to simplify the given expression.
(a - b)3 = a3 - 3a2b + 3ab2 - b3
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(p - q)3 - (p + q)3 :
= (p3 - 3p2q + 3pq2 - q3) - (p3 + 3p2q + 3pq2 + q3)
= p3 - 3p2q + 3pq2 - q3 - p3 - 3p2q - 3pq2 - q3
= -3p2q - q3 - 3p2q - q3
= -6p2q - 2q3
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