Avon High School
Section: 1.6
ACE COLLEGE ALGEBRA II - NOTES
Other Types of Equations
Mr. Record: Room ALC-129
Day 1 of 1
Polynomial Equations
A polynomial equation is the result of setting two polynomials equal to each other. The equation is in general
form if one side is 0 and the polynomial on the other side in descending powers of the variable. The degree of
a polynomial equation is the same as the highest degree of any term in the equation.
Example 1
Solving Polynomial Equations by Factoring
Solve by factoring.
a. 4 x 4  12 x 2
b. 2 x3  3x 2  8 x  12
Investigation: Use a graphing calculate to sketch the general form of
each problem in Example 1. Where do you see the solutions?
Radical Equations
A radical equation is an equation in which the variable occurs in a square root, cube root, or any higher root.
Solving Radical Equations Containing nth Roots
1.
2.
3.
4.
If necessary, arrange terms so that one radical is isolated on one side of the equation.
Raise both sides of the equation to the nth power to eliminate the isolated nth root.
Solve the resulting equation. If this equation still contains radicals, repeat steps 1 and 2.
Check all proposed solutions in the original equation.
Example 2
Solving a Radical Equation
Solve
x3 3 x
Solving an Equation That Has Two Radicals
Solve x  5  x  3  2 .
Example 3
Equations with Rational Exponents
m
Recall that a n 
 a
n
m
 n am .
3
For example 3 4 x3  6  0 can be expressed as 3x 4  6  0.
Solving Radical Equations of the Form x m/n = k.
m
is in lowest terms, and k is a real number.
n
1. Isolate the expression with the rational exponent.
Assume that m and n are positive integers,
2. Raise both sides of the equation to a power that is the reciprocal of the rational exponent, namely,
If m is even:
m
n
If m is odd:
m
x k
xn  k
n
 mn  mn
m
x
k


k
 
 
n
 mn  mn
m
x
k

k
 
 
n
m
n
m
x  k
n
.
m
xk
It is incorrect to insert the ± symbol when the numerator of the exponent is odd. An odd index has only
one root.
3. Check all proposed solutions in the original equation to find out if they are actual solutions or
extraneous solutions.
Example 4
Solving Equations Involving Rational Exponents
3
2
a. Solve 5 x  25  0
2
b. Solve x 3  8  4
Equations That Are Quadratic in Form
An equation that is quadratic in form is one that can be expressed as a quadratic equation using an appropriate
substitution. Look at the chart below.
Given Equation
Substitution
New Equation
4
2
x  10 x  9  0
or
u 2  10u  9  0
u  x2
2 2
2
 x   10 x  9  0
1
2
3
5 x  11x 3  2  0
1
or
2
1
 1
5  x 3   11x 3  2  0
 
Example 5
u  x3
5u 2  11u  2  0
Solving Equations in Quadratic Form
a. Solve x 4  5 x 2  6  0 .
b. Solve x 4  5 x 2  6  0 .
Equations Involving Absolute Value
Example 6
Solving an Equation Involving Absolute Value
a. Solve 2 x 1  5 .
b. Solve 4 1  2 x  20  0 .
Applications
Example 7
Commercial Clutter
By 2005, the amount of “clutter,” including commercials and plugs
for other shows had increased to the point where an “hour-long”
drama on cable TV was 45.4 minutes. The graph to the right shows
the average number of nonprogram minutes in an hour of prime-time
cable television. Although the minute of clutter grew from 1996
through 2005, the growth was leveling off. The data can be modeled
by the formula
M  0.7 x  12.5
where M is the average number of nonprogram minutes in an hour of
prime-time cable x years after 1996. Assuming the trend from 1996
to 2005 continues, use the model to project when there will be 16
cluttered minutes in every prime-time cable TV hour.