
Algebraic
Expressions 


Algebraic expressions  preliminaries 
Evaluating algebraic
expressions 
Simplifying algebraic
expressions 
Expanding
algebraic expression by removing parentheses (i.e. brackets) 
The square of a
binomial  the perfect square trinomial 
The square of a trinomial 





Algebraic expressions  preliminaries

An
algebraic expression is one or more algebraic terms
containing variables and constants connected by mathematical
operations. Terms
are the elements separated by the plus or minus signs.

In
algebraic expressions, variables are letters, such as
a,
b, c,
or x,
y,
z,
that can have different values.

Constants
are the terms or elements represented only by numbers. Coefficients
are the number part of the terms that multiply a variable or
powers of a variable.

An
algebraic expression consisting of a single term is called a monomial, expression consisting of two terms is
binomial, three
terms trinomial and an expression with more than three terms is
called polynomial.


Evaluating algebraic expressions 
To
evaluate an algebraic expression means to replace (substitute)
the variables in the expression with numeric values that are
assigned to them and perform the operations in the expression. 

Example: 

Evaluate
the expression: 1
+
a^{2}b
+ 1/4a^{4}b^{2 }
for a
= 1 and
b = ^{ }2. 
Solution: 

1 + (1)^{2
}·^{
}(2)
+ 1/4 ·^{ }(1)^{4
}·^{
}(2)^{2}
= 1  2
+ 1/4
·^{ }1
·
4 = 0. 


Or,
since 1+
a^{2}b
+ 1/4a^{4}b^{2 }
= (1+ 1/2a^{2}b)^{2} 


^{ }then,
by substituting a
= 1 and
b
= ^{ }2,
[1 + 1/2·(1)^{2
}·^{ }(2)]^{2
}= 0. 


Simplifying algebraic
expressions 
By
simplifying an algebraic expression, we mean reducing it in the
simplest possible form which mainly involves: multiplication
and division, removing (expanding) brackets and collecting (adding
and subtracting) like
terms. 
Like
terms are those terms which contain the same powers of same
variables and which can only differ in coefficients. 
Examples: 

a)
 4a^{3
}+
3a^{2
}+
5a^{3
} 7a^{2
} = ( 4
+
5) · a^{3
}+
(3  7)
· a^{2 } = a^{3 } 4a^{2}, 


b)
(x^{2
} x^{
}+
1) ·
(x +
1) =
x^{3 } x^{2}
+
x
+
x^{2} ^{
} x^{
}+
1 = x^{3}
+
1. 


Expanding
algebraic expression by removing parentheses (i.e. brackets) 
The
operation of multiplying out algebraic expressions that involve
parentheses using the distributive property is often
described as expanding the brackets. 
Some
important binomial products like perfect squares, and difference
of two squares are used to help with factoring algebraic
expressions. 

Examples: 

a)
(a
 b)^{2}
= (a
 b)
· (a
 b) =
a^{2
} ab^{
} ab^{
}+
b^{2}
= a^{2
} 2ab^{
}+
b^{2}, 


b)
(a
 b)
·
(a
+
b) =
a^{2 } ab^{
}+
ab^{
} b^{2} =
a^{2 }
b^{2}, 


c)
(x^{
}+
y)
· (x^{2}
 xy
+
y^{2}) =
x^{3 } x^{2}y
+
xy^{2}
+
x^{2}y ^{
} xy^{2
}+
y^{3 }
= x^{3}
+
y^{3}. 


The
square of a binomial (or binomial square ) 
To the
square of the first term add twice the product of the two terms
and the square of the last term. 
Examples: 

a)
(a +
b)^{2}
= (a
+ b)
·
(a
+ b) =
a^{2 }+
ab^{
}+ ab^{
}+
b^{2}
= a^{2
}+ 2ab^{
}+
b^{2}, 


b)
(2x +
3)^{2} =
(2x)^{2}
+ 2
· (2x) ·
3
+
3^{2}
= 4x^{2} +
12x +
9, 


c)
(x^{
} 2y)^{2} =
x^{2} +
2 · x
·
(2y)
+
(2y)^{2} =
x^{2} 
4xy +
4y^{2}. 


Squaring
trinomial or trinomial square 
To the sum of squares of the 1st, the 2nd and the 3rd term add, twice the product of the 1st and the 2nd term, twice the product of the 1st and the 3rd term, and twice the product of the 2nd and the 3rd term. 

Examples:


a) (x^{2
}  2x
+ 5)^{2}
= (x^{2})^{2
}+ (2x)^{2}
+ 5^{2}
+ 2
·
x^{2 }· (2x)^{
}+ 2
·
x^{2 }· 5
+
2^{ }·
(2x)^{
}·
5
= 


= x^{4
}+ 4x^{2}
+ 25
 4x^{3
}+ 10x^{2}
 20x
= x^{4 }  4x^{3
}+ 14x^{2}
 20x
+ 25, 



b) (a^{3
}
a^{2}b  3ab^{2})^{2}
= (a^{3})^{2 }+ (a^{2}b)^{2}
+ (3ab^{2})^{2}
+ 2a^{3
}(a^{2}b)^{
}+ 2a^{3
}(3ab^{2})
+ 2(a^{2}b)^{
}(3ab^{2})
= 


= a^{6
}+ a^{4}b^{2
}+ 9a^{2}b^{4
} 2a^{5}b
 6a^{4}b^{2
}+ 6a^{3}b^{3
}= a^{6}  5a^{4}b^{2
}+ 9a^{2}b^{4
} 2a^{5}b^{
}+ 6a^{3}b^{3}. 










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