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 x2 is 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.
For any real numbers a, b 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
For any real numbers a, b and c,
a + b = b + a
a ⋅ b = b ⋅ a
Examples :
5 + 8 = 8 + 5
5 ⋅ 8 = 8 ⋅ 5
For any real numbers a, b 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)
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,
= (2y2 + y2)/3
= 3y2/3
= y2
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