# COMBINING LIKE TERMS WORKSHEET

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 + 3a- 5a - 2) and (a- 6a+ 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 :

(x- 3x- 2x+ x2) + 5(3x+ 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(5x- 2x + 8).

Problem 21 :

Subtract 2(y2 + 3y - 1) from 5(2y- 2y- 7y - 4).

Problem 22 :

Simplify :

(x+ y2) - (x + y)(x - y)

Problem 23 :

Simplify :

(p - q)3 - (p + q)3

= (3x2 + 7x - 2) + (x2 -  4x + 5)

= 3x2 + 7x - 2 + x2 -  4x + 5

Group the like terms together.

= (3x+ x2) + (7x - 4x)  + (-2 + 5)

Combine like terms.

= 4x2 + 3x + 3

= (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

= (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

= 7ab - 3ab

Both the terms in the above expression are like terms. So, we can combine them.

= 4ab

= (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

= (a3 + 3a- 5a - 2) + (a- 6a+ 9a - 8)

= a3 + 3a- 5a - 2 + a- 6a+ 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

= (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

= (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

= (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

= (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

= 6y- 21y4 + y3 + 2y+ (-3y + 6y) + (1 - 15)

= 6y- 21y4 + y3 + 2y+ 3y + (-14)

= 6y- 21y4 + y3 + 2y+ 3y - 14

= -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

= 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

= -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

= 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

= 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

= (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

= (x- 3x- 2x+ x2) + 5(3x+ 15x7 + 4x6 + 2x3 + 6x2)

Distributive Property.

= x- 3x- 2x+ x2 + 15x+ 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

= (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

= (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

= 3(5x- 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) + 5x+ (-6x - 7x) + (24 + 3)

= 13x3 + 5x+ (-13x) + 27

= 13x3 + 5x- 13x + 27

= 5(2y- 2y- 7y - 4) - 2(y2 + 3y - 1)

Using Distributive Property,

= 5(2y- 2y- 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

= (x+ y2) - (x + y)(x - y)

= x+ y2 - [(x + y)(x - y)]

= x+ y2 - [x2 - xy  + xy - y2]

= x+ y2 - [x2 - y2]

= x+ y2 - x2 + y2

= 2y2

The following algebraic identities can be used to simplify the given expression.

(a - b)= a3 - 3a2b + 3ab2 - b3

(a + b)= 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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