Assignment 1 ± Polynomial Equations
An equation is a statement of equality between two expressions, and may involve one or more variables.
A particular choice of value for the variable(s) which makes an equation true (i.e. the expression on the left hand
side have a value equal to that of the expression on the right hand side) is called a solution or root of the equation;
the collection of all solutions is called the solution set.
±
x
+ 6 = 15
has solution
x
= 9
(i.e., its solution set is
f
9
g
)
±
the equation
x
2
= 25
has two solutions, namely
x
=
±
5
(i.e., its solution set is
f²
5
;
5
g
)
±
thus, the equation
x
2
=
²
1
has no real solution (i.e., its solution set is
?
)
Solving Polynomial Equations (of one variable)
Isolating the variable using "opposite" operations:
If all occurrences of the variable have the same power, then we can solve the equation by isolating the variable.
We do this by applying the following principles.
±
Addition Principle:
Adding an expression which is defined everywhere to both sides of an equation yields an
equivalent [ i.e., has exactly the same solution set ] equation. Wherever the added expression is undefined,
the original equation must be checked.
a
=
b
,
a
+
c
=
b
+
c
, for all
c
2
R
±
Multiplication Principle:
Multiplying an equation by an expression that is defined everywhere and never
equal to zero yields an equivalent equation. Wherever the new expression is
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 Fall '10
 HU
 Equations, Inequalities, Quadratic equation, Elementary algebra, real solutions

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