Common Algebra Errors
The following is a list of some common algebraic errors.
Order of Operations
Remember that when simplifying an algebraic expression, the order in which operations
are performed is important.
 Do things in Parentheses first
 Exponents
 Multiplication and Division from left to right
 Addition and Subtraction from left to right.

ab  c   ab  ac 
Exponents
  4 2  16 the order of operations says to square the 4 first, then apply the minus
sign to the result
n
 Raising a binomial to a power: a  b  a n  b n
For example, squaring a binomial: a  b  a 2  b 2
You need to use FOIL: a  ba  b  a 2  2ab  b 2
2
a  b3  a 3  3a 2b  3ab 2  b3









a 
n m
Review the binomial theorem
 a nm It equals a nm
1
1
2
It equals x 2

x
2
x
4
1
 4 x 2 It equals 4 x 2
1
x 2
5
5
5
It
equals

1
1
3x  2 3x  2 2
3x  2 2
1
 5 x It equals 5 x
2
2
4x  1  4 x  4 It equals 4 x 2  2 x  1
5
5
2
 53x  It equals x  2
2
3
3x
1
sin 1 x 
It equals arcsin x
sin x
2
sin 2 x  sin x 2 It equals sin x 
5x
2


Page 1
Common Algebra Errors
Fractions
 y1 1 x x y
1 1
1

You need to first find the LCM:      
 
xy
x y x y
 y x y x
3
3
 3
Does this mean x or
? Do not write fractions with a diagonal slash.
5x
5
5x







a  b 1 b
b
or 
Only common factors cancel.

a  c 1 c
c
3x 2  2 x  1 3  2 x  1
The x 2 factors do not cancel out. Instead factor the

2
2  2x
2x  2x
3x  1( x  1) and cancel common factors
numerator and denominator:
2 x( x  1)
6
4
To simplify x
, don’t multiply by x. We can only multiply an expression by
5x
some form of 1, otherwise the value of the fraction changes. Instead simplify in
6
4
 x  6  4x
x
the following way:
  
5x  x 
5x 2
4x  3
3
 2 x  3 instead it equals 2 x 
2
2
2
x 1
x 2  1 x  1( x  1)
Need to factor numerator and simplify:

 x 1
x 1
x 1
( x  1)
4
4 4

 can’t write as two separate fractions
5x  3 5x 3
If the fraction is reversed, so there is only 1 term in the denominator, then you can
5x  3 5x 3
split up the fraction:


4
4 4
2
x  2   4 
When dividing one fraction by another, multiply by the
4
x  3x  5 
3x  5
2  3x  5  3x  5
reciprocal of the second fraction:


x 4 
2x
Simplifying Expressions
 4 x  2  6 x You can only add like terms
 3x 2  5x  4  2 x 2  6 x  10  3x 2  5x  4  2 x 2  6 x  10
You need to remember to subtract all terms in the second parentheses.
3x 2  5x  4  2 x 2  6 x  10  3x 2  5x  4  2 x 2  6 x  10

 


 

Page 2
Common Algebra Errors
Solving Equations
 4 x 2  2 x is not equivalent to 2 x  1 . Cancelling out the x is only correct if x is not
zero. Instead rewrite equation as: 4 x 2  2 x  0  2 x(2 x  1)  0 and solve using
the zero-product rule.
Inequalities
 x  y so kx  ky where k is a constant. This is only true when k  0 . If k  0,
then kx  ky .
Functions
Assuming all functions are linear functions
 f ( x  y)  f ( x)  f ( y) and f (ax)  af ( x) Unless f (x) is a linear function

5 x 2  3x  5 x 2  3x
1
1 1
 
x4 x 4
log( x  y)  log( x)  log( y)



log x  log x
sin(2 x)  2 sin( x)
sin( x  y)  sin( x)  sin( y)


Trig functions
 sin( x)  sin

Trig functions always require an argument.
sin 2 x
 sin 2 In the term sin 2x, 2x is the argument of sine, it does not mean
x
sin 2 multiplied by x. It is helpful to write sin 2 x as sin 2 x  to emphasize the
argument of the sine function.
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