Download College Algebra Lecture Notes, Section 1.6

College Algebra Lecture Notes
Section 1.6
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Section 1.6: Solving Other Types of Equations
Big Idea: .
Big Skill: .
A. Polynomial Equations of Higher Degree
 A polynomial equation of degree n is an equation of the form:
a0  a1 x  a2 x 2  a3 x 3   an x n  0
 Polynomials are the most important topic in algebra because any equation that can be
written using addition, subtraction, multiplication, division, integer powers, or roots
(which are rational powers) can be solved by converting the equation into a polynomial
equation.
x2  1
2
 For example, the equation
  x  2  can be transformed into the polynomial
x3
5
4
2
equation x  6 x  38 x  80 x  47  0 . Both of these equations have the same solutions,
one of which is x = 1, as is shown in the graph below:
Steps for solving any polynomial equation (degree 2 or higher):
 Expand the polynomial equation (if needed), and collect all terms on one side; combine
like terms.
 Factor the polynomial on the one side.
 Set each factor equal to zero (this is justified by the zero-product property rule).
 Solve each first degree equation for the variable.
 Check answers in original equation.
 NOTE: The degree of the polynomial tells how many solutions there must be.
Practice:
1. Solve x3  13x 2  42 x .
College Algebra Lecture Notes
Section 1.6
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College Algebra Lecture Notes
2. Solve 7 x 2  15 x  2 x 3 .
3. Solve x3  18  9 x  2 x 2 .
4. Solve x 6  64  0 .
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College Algebra Lecture Notes
Section 1.6
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B. Rational Equations
Solving Rational Equations (Book’s Method)
 Identify and exclude any values that cause a zero denominator.
 Multiply both sides by the LCD and simplify.
 Solve the resulting equation.
 Check all solutions in the original equation.
Solving Rational Equations (Another Method)
 Get all terms on one side of the equation.
 Write all terms as equivalent fractions with a common denominator.
 Add the terms.
 Factor the numerator and cancel any common factors in the denominator.
 Set the numerator to zero and solve the resulting “mini equations”.
 Check all solutions in the original equation.
Practice:
5. Solve
2
1
4

 2
.
m m 1 m  m
College Algebra Lecture Notes
6. Solve x 
Section 1.6
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12
4x
 1
.
x3
x3
C. Radical Equations and Equations with Rational Exponents
Solving a radical equation with one radical expression:
 Isolate the radical expression (i.e., get it on one side of the equation by itself with all
other terms on the other side).
 Raise both sides of the equation to the power of the index of the radical.
 Solve the resulting equation.
 Check your answer. It is common to get answers that are not truly solutions. These
answers are called extraneous solutions.
Solving a radical equation with two radical expressions:
 Isolate ONE of the radical expressions.
 Raise both sides of the equation to the power of the index of the radical to eliminate the
first radical.
 Solve the resulting radical equation (that now only has one radical).
 Check your answer. Eliminate any extraneous solutions.
The Power Property of Equality:
If n u  v ,
Then
 u
n
n
 vn
Which simplifies to u  v n
Practice:
7. Solve
5 x  6  3  2 .
College Algebra Lecture Notes
3x  11  x  5 .
8. Solve
9. Solve
10. Solve
3
2x  3  5  8 .
x  15  x  3  2 .
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College Algebra Lecture Notes
D. Equations in Quadratic Form
Practice:
2
1
11. Solve x 3  3x 3  10  0 .
12. Solve x 4  36  5 x 2 .
E. Applications
Practice:
13. Hmmm...
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