# SIMPLIFYING ALGEBRAIC EXPRESSIONS

A term is a number, a variable, or a product or quotient of numbers and variables. For example 5, x, 7a, b2, and 2m3n are all terms. Like terms contain identical variables. For example, in 5x2 - 3x2 + 3x, the terms 5x2 and -3x2 are like terms because the variable part of each term is identical.

The coefficient of a term is a number that multiplies a variable. For example, in 10x2, the coefficient of xis 10, and in 4m/5, the coefficient m is 4/5.

An expression is in simplest form when it is replaced by an equivalent expression having no like terms or parentheses. Simplifying means rewriting in simpler form.

## Distributive Property

For any real numbers ab and c,

a(b + c) = ab + ac

a(b - c) = ab - ac

Examples :

5(2 + 7) = 5 ⋅ 2 + 5 ⋅ 7

5(7 - 3) = 4 ⋅ 7 - 4 ⋅ 3

## Commutative Property

For any real numbers ab and c,

a + b = b + a

⋅ b = b ⋅ a

Examples :

5 + 8 = 8 + 5

⋅ 8 = 8 ⋅ 5

## Associative Property

For any real numbers ab and c,

(a + b) + c = a + (b + c)

(ab)c = a(bc)

Examples :

(2 + 3) + 4 = 2 + (3 + 4)

(2 ⋅ 3⋅ 4 = 2 ⋅ (3 ⋅ 4)

## Solved Problems

Simplify the following algebraic expressions :

Problem 1 :

4x3 - 2(x3 + 3x)

Solution :

= 4x3 - 2(x3 + 3x)

Using Distributive Property,

= 4x3 - 2x3 - 2x

Combining like terms,

= 2x3 - 2x

Problem 2 :

5.4(x - 2y) - 2.7(x - 3y)

Solution :

= 5.4(x - 2y) - 2.7(x - 3y)

Using Distributive Property,

= 5.4x - 10.8y - 2.7x + 8.1y

Combining like terms,

= 2.7x - 2.7y

Problem 3 :

(1/2)(2a + 3b + 4c) - (3/2)(b + 2c)

Solution :

= (1/2)(2a + 3b + 4c) - (3/2)(b + 2c)

Using Distributive Property,

= a + 3b/2 + 2c - 3b/2 - 3c

Combining like terms,

= a - c

Problem 4 :

a(b - c) - b(a + c) - c(a - b)

Solution :

= a(b - c) - b(a + c) - c(a - b)

Using Distributive Property,

= ab - ac - ab - bc - ac + bc

Combining like terms,

-2ac

Problem 5 :

3[6a - 3(1 - a) - 5(a + 1)]

Solution :

=  3[6a - 3(1 - a) - 5(a + 1)]

Using Distributive Property,

=  3[6a - 3 + 3a - 5a - 5]

Combining like terms,

=  3[4a - 8]

Using Distributive Property,

=  12a - 24

Problem 6 :

(2/3)(y2 - y - 3) + (1/3)(y2 + 2y + 6)

Solution :

=  (2/3)(y2 - y - 3) + (1/3)(y2 + 2y + 6)

Using Distributive Property,

=  2y2/3 - 2y/3 - 6/3 + y2/3 + 2y/3 + 6/3

=  2y2/3 + y2/3 - 2y/3 + 2y/3 - 6/3 + 6/3

Combining like terms,

=  (2y+ y2)/3

=  3y2/3

=  y2

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