Hawkes Learning Systems: College Algebra
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Transcript Hawkes Learning Systems: College Algebra
HAWKES LEARNING SYSTEMS
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Copyright © 2010 Hawkes Learning Systems.
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Hawkes Learning Systems:
College Algebra
Section 2.4: Higher Degree Polynomial
Equations
HAWKES LEARNING SYSTEMS
Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Objectives
o Solving quadratic-like equations.
o Solving general polynomial equations by factoring.
o Solving polynomial-like equations by factoring.
HAWKES LEARNING SYSTEMS
Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
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Solving Quadratic-Like Equations
o A polynomial equation of degree n in one variable,
say x , is an equation that can be written in the form
a n x a n 1 x
n
n 1
... a1 x a 0 0
where a i is a constant and a n 0 .
o In general, there is no method for solving polynomial
equations that is guaranteed to find all solutions.
o Since the Zero-Factor property applies whenever a
product of any finite number of factors is equal to 0,
we can use this property to solve quadratic-like
equations.
HAWKES LEARNING SYSTEMS
Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
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Solving Quadratic-Like Equations
An equation is quadratic-like, or quadratic in
form, if it can be written in the form
aA bA c 0
2
Where a , b , and c are constants, a 0 , and A is
an algebraic expression. Such equations can be
solved by first solving for A and then solving for
the variable in the expression A .
HAWKES LEARNING SYSTEMS
Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
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Example 1: Solving Quadratic-Like Equations
Solve the quadratic-like equation.
x
Step 1: Let
3x
2
2
2
2
A 4 A 2 0
and factor.
x 3x
2
and solve
for x .
2
A 2A 8 0
A x 3x
Step 2: Replace
A with
2 x 3x 8 0
A4
or
A 2
x 3x 4
or
x 3 x 2
x 3x 4 0
or
x 3x 2 0
x 4 x 1 0
or
x 2 x 1 0
x 4 or x 1
or
2
2
2
2
x 2 or x 1
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Example 2: Solving Quadratic-Like Equations
Solve the quadratic-like equation.
2
1
x 3 5x3 6 0
2
3
13
x 5 x 6 0
1
1
1
3
3
x 1 x 6 0
1
1
x 3 1
x ( 1)
x 1
3
or
x3 6
or
or
x6
3
x 216
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Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
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Solving General Polynomial Equations by
Factoring
o If an equation of the following type can be factored
completely, then the equation can be solved by using
the Zero-Factor Property.
a n x a n 1 x
n
n 1
... a1 x a 0 0
o If the coefficients in the polynomial are all real, the
polynomial can, in principle, be factored.
o In practice, this may be difficult to accomplish unless
the degree of the polynomial is small or the
polynomial is easily recognizable as a special
product.
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All rights reserved.
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Example 3: General Polynomial Equations
Solve the equation by factoring.
4
x 40
4
Step 1: Isolate
0 on one
side and
factor.
Step 2: Set both
equations
equal to 0
and solve.
x 4
x
x 20
2
x 2
2
x i 2
2
2 x 2 0
2
or
or
or
x 20
2
x 2
2
x 2
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Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 4: General Polynomial Equations
Solve the equation by factoring.
z z 9z 9 0
3
z
2
2
z 1 9 z 1 0
z 1 z 9 0
2
z 1 z 3 z 3 0
z 1 0
or z 3 0
z 1 or
z3
or z 3 0
or
z 3
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Copyright © 2010 Hawkes Learning Systems.
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math courseware specialists
Example 5: General Polynomial Equations
Solve the equation by factoring.
64 t 8 0
3
(4 t ) 2 0
3
3
4 t 2 16 t 2 8 t 4 0
4t 2t 1 0
2
4t 2 0
4t 2
t
1
2
or
or
or
t
t
2 4 4 1
2
2
24
2
12
8
t
1 i 3
4
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Copyright © 2010 Hawkes Learning Systems.
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Solving Polynomial-Like Equations by Factoring
o Some equations that are not polynomial can be
solved using the methods we have developed in
Section 2.4.
o The general idea will be to rewrite the equation so
that 0 appears on one side, and then to apply the
Zero-Factor Property.
HAWKES LEARNING SYSTEMS
Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 6: Polynomial-Like Equations
Solve the following equation by factoring.
11
6
1
x 5 3x 5 4x 5
11
Step 1: Isolate 0 on
one side.
6
1
x 5 3x 5 4 x 5 0
1
Step 2: Factor.
x 5 x 3x 4 0
2
1
x 5 x 4 x 1 0
Step 3: Apply the
Zero-Factor
Property.
1
x5 0
x0
x40
x 4
x 1 0
x 1
HAWKES LEARNING SYSTEMS
Copyright © 2010 Hawkes Learning Systems.
All rights reserved.
math courseware specialists
Example 7: Polynomial-Like Equations
Solve the following equation by factoring.
x 3
x 3
1
2
1
2
1
2 x 32 0
1 2 x 3 0
1
x 3 2 2 x 5 0
x 3
1
2
0
or
2x 5 0
1
0
or
2x 5
1
x 32
No Solution
x
5
2