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Simplifying Algebraic Expressions
The Order of Operations:
Please – parentheses and other grouping symbols - work from the inside out!
Parentheses ( )
Brackets [ ]
Braces { }
The fraction bar
Excuse – exponents. Examples of exponential expressions:
2  2 2 2  42  8
3
3   1 3  3   3  3   9
2
(3)  (3)(3)   9
2
Base is 2
Exponent is 3
Base is 3
Exponent is 2
Coefficient of 32 is -1
Base is -3
Exponent is 2
My Dear – multiply and divide, in order, from left to right
Aunt Sally – add and subtract, in order, from left to right
Equation: two algebraic expressions with an equal sign between them
Formula: an equation that uses variables to express a relationship between two or
more quantities.
Mathematical Modeling: The process of finding formulas to describe real-world
phenomena
Terms: the parts of an algebraic expression that are separated by addition or
subtraction signs that are not inside parentheses.
Constant: a term with just a number (also called numerical term)
Variable: a letter that represents a variety of different numbers
Algebraic Expression: a combination of variables and numbers using the operations of
addition, subtraction, multiplication, or division, as well as powers or roots
Simplifying Algebraic Expressions
Evaluating an Algebraic Expression: finding the value of the expression for a given
value of the variable We accomplish this by using substitution. To avoid careless sign
mistakes, use parentheses around the number that you are substituting.
Factors: The parts of a term connected by multiplication. In 5x the factors are 5 and x.
In 3(x + 2) the factors are 3 and (x + 2)
Coefficient: The numerical part of a term - coefficients of 1 and -1 are not written.
5x
x
−x
Coefficient of x is 5
Coefficient of x is 1 (invisible)
Coefficient of x is −1
Like terms: Terms with the same variable factors
5x and 8x
y and −4y
5x2 and 7x2
3xyz and −2xyz
Note: 2xy and 5yx are like terms (the order of the factors doesn’t matter)
Properties of Real Numbers:
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Distributive Property (with addition):
Distributive Property (with subtraction):
a+b=b+a
a*b=b*a
a + (b + c) = (a + b) + c
a * (b * c) = (a * b) * c
4( x + 2) = 4(x) + 4(2) = 4x + 8
4( x - 2) = 4(x) - 4(2) = 4x - 8
Note: − ( x – 3) = −1( x – 3) = −x + 3
9 – ( x + 5 ) = 9 – x – 5 You must multiply (distribute) before you combine
–x +4
Combining like terms: Using the properties of real numbers to simplify an expression
containing like terms. An algebraic expression is simplified when parentheses have
been removed and like terms have been combined.