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Section 1.6 Other Types of Equations
*Polynomial Equations
A polynomial equation in x is a equation of a form
a n x n  a n 1 x n 1    a 2 x 2  a1 x  a0  0 (called General Form)
where ai ' s are real numbers.
The degree of a polynomial equation is the highest degree of any term in the equation.
For example, x 3  x 2  4 x  4 is a polynomial equation of degree 3. In particular, a
polynomial equation of degree 1 is a linear equation and a polynomial equation of
degree 2 is a quadratic equation.
Example 1) Solve by factoring: 4 x 4  12 x 2
Step 1 Move all terms to one side and obtain zero on the other side.
Step 2 Factor.
Step 3 Set each factor equal to zero and solve the resulting equation (The Zero-Product
Principle).
Step 4 Check the solutions in the original equation.
Example 2) Solve by factoring: 2 x 3  3x 2  8 x  12
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*Radical Equations
A radical equation is an equation which the variable occurs in a square root, cube root,
or any higher root.
Example 3) Solve: x  3  3  x
Step 1 Isolate a radical on one side.
Step 2 Raise both sides to the nth power.
Step 3 Solve the resulting equation.
Step 4 CHECK the solution in the original equation.
Example 4) Solve: x  5  x  3  2 .
If an equation has two or more radicals, repeat Step 1 and Step 2 until you get a
polynomial equation.
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*Equations with Rational Exponents
m
Recall that a n  (n a ) m  n a m .
m
Solving Radical Equation of the Form x n  k .
1. Isolate the expression with the rational exponent.
n
2. Raise both sides of the equation to the
power that is the reciprocal of the rational
m
exponent.
If m is even:
If m is odd:
m
m
xn k
xn k
n
n
n
 mn  m
 x   k m
 
 
n
 mn  m
x   km
 
 
n
n
x  k m
x km
3. CHECK the solutions in the original equation.
Example 5) Solve:
3
2
a. 5 x  25  0
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b. x 3  8  4
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*Equation That Are Quadratic in Form
An equation that can be expressed as a quadratic equation can be solved using an
appropriate substitution.
Example 6) Solve: x 4  5 x 2  6  0
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Example 7) Solve: 3 x 3  11x 3  4  0
*Equation Involving Absolute Value
If c is a positive real number and X represents any algebraic expression, then
X  c is equivalent to X  c or X  c .
Example 8) Solve: 2 x  3  11
Example 9) Solve: 41  2 x  20  0
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