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Hawkes Learning Systems:
College Algebra
Section 5.1: Introduction to Polynomial
Equations and Graphs
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Objectives
o Zeros of polynomials and solutions of polynomial
equations.
o Graphing factored polynomials.
o Solving polynomial inequalities.
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Zeros of a Polynomial
The number k is said to be a zero of the
polynomial function
f  x   an x n  an1 x n1  ...  a1x  a0
if f  k   0. This is also expressed by saying
that k is a root or a solution of the equation
f  x   0.
Note: k may be a complex number.
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Zeros of a Polynomial
If f is a polynomial with real coefficients and if k is a
real number zero of f, then the statement f  k   0
means the graph of f crosses the x-axis at  k ,0  .
In this case,  k ,0 may be referred to as an x-intercept
of f. .
y
f
 k ,0 
x
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Polynomial Equations
A polynomial equation in one variable, say
the variable x, is an equation that can be
written in the form
an xn  an1xn1  ...  a1x  a0  0
where an , an1 ,..., a1 , a0 are constants.
Assuming an  0, we say such an equation is
of degree n.
For example: 6 x 2  3 x  1  0 or x 3  7  0.
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Example 1: Zeros of Polynomials and Solutions
of Polynomial Equations
Verify that the given value of x solves the corresponding
polynomial equation.
2
2  x3  x 2   12 x; x  
2
2

  
3
2
  12
?
2  8  4  24
?

Substitute –2 for x in the original
equation.
Simplify, and solve the equation.
24  24
Thus, x  2 is a solution to the equation.
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Example 2: Zeros of Polynomials and Solutions
of Polynomial Equations
Verify that the given value of x solves the corresponding
polynomial equation.
3i 7
2
3x  1  4 x ; x 
8
2
4 x  3x  1  0
Although we could verify the
solution by substituting for x,
it is easier to solve this
equation for ourselves using
the quadratic formula.
x
  3 
 3
2  4
2
 4  41
3i 7
x
8
Continued on the next slide…
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Example 2: Zeros of Polynomials and Solutions
of Polynomial Equations (Cont.)
3i 7
3x  1  4 x ; x 
8
2
3i 7
x
8
One of the two resulting solutions for x is equivalent to
the value we were given for x at the beginning of the
problem, and thus the given value of x solves the
equation.
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Example 3: Zeros of Polynomials and Solutions
of Polynomial Equations
Verify that the given value of x solves the corresponding
polynomial equation.
2
x
5 x3 
; x0
2i
5  0 
3 ?
 0
2
2i
00
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Graphing Factored Polynomials
The behavior of a polynomial function as x   can be
determined as follows:
o As x  , the leading term of
.f  x   an x n  an1 x n1  ...  a1x  a0 dominates the behavior.
o If n is even, x n   as x   , and if n is odd, then
.x n   as x   and x n   as x  .
o If an is positive, multiplying x n by an merely compresses
or stretches the graph of x n, while if an is negative, the
graph of an x n is the reflection with respect to the x-axis
n
of the graph of an x .
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Graphing Factored Polynomials
Summary:
n even
n odd
x 
xn  
xn  
x  
xn  
x n  
an positive
No change.
an negative
an x n is reflected
over the x-axis.
Note: an stretches or compresses the graph.
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Graphing Factored Polynomials
n
n 1
f
x

a
x

a
x
 ...  a1x  a0 the y-intercept is
For   n
n 1
 0, a0  .
y
f  x
 0, a0 
x
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Graphing Factored Polynomials
If we are able to factor a given polynomial f into a product
of linear factors, every linear factor with real coefficients
will correspond to an x-intercept of the graph of f. For
example,  3x  5 x  2 2 x  6   0 has the x-intercepts:
y
5 
f  x
x   2,0 ,  ,0 ,  3,0 .
3 
 3,0 
 2,0 
5 
 ,0
3 
x
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Example 4: Graphing Factored Polynomials
Sketch the graph of the following polynomial function, paying
particular attention to the x-intercept(s), the y-intercept, and the
behavior as x  .
f  x    x  2 x  1 x  2 
 1 
x-intercepts:   , 0 ,  0, 0 ,  2, 0
 2 
y -intercept:  0, 0
If we were to multiply out the three linear factors of f, the highest
degree term would be 2x3. The degree of f and the fact that the
leading coefficient is negative indicates how f behaves as x  .
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Example 5: Graphing Factored Polynomials
Sketch the graph of the following polynomial function, paying
particular attention to the x-intercept(s), the y-intercept, and the
behavior as x  .
g  x   x4 1
g  x    x  1 x  1  x 2  1
x-intercepts: 1, 0 ,  1, 0
y -intercept:  0, 1
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Example 6: Graphing Factored Polynomials
Sketch the graph of the following polynomial function, paying
particular attention to the x-intercept(s), the y-intercept, and the
behavior as x  .
h  x   x2  2x  3
h  x    x  3 x  1
x-intercepts:  3,0 , 1,0
y -intercept:  0, 3
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Solving Polynomial Inequalities
Every polynomial inequality can be rewritten in the
form f  x   0, f  x   0, f  x   0, or f  x   0,
where f is a polynomial function. This will be the key
to solving the inequality.
By graphing the polynomial f, we will be able to easily
pick out the intervals that solve the inequality.
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Example 7: Solving Polynomial Inequalities
Solve the following polynomial inequality.
2 x 2  3x  9
2 x 2  3x  9  0
 2 x  3 x  3  0
Now graph the function
f  x   2 x  3 x  3 using:
 3 
x-intercepts:   ,0 ,  3,0
 2 
y-intercept:  0, 9
3

 ,     3,  
2

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Example 8: Solving Polynomial Inequalities
Solve the following polynomial inequality.
x 4  2 x 2   x3
x 4  x3  2 x 2  0
x2  x2  x  2  0
x 2  x  2  x  1  0
Now graph the function
2
f  x   x  x  2 x  1 using:
x-intercepts:  2,0 ,  0,0 , 1,0
y-intercept:  0,0
 2,0   0,1
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Example 9: Solving Polynomial Inequalities
Solve the following polynomial inequality.
 x  3 x  1 x  2  0
Graph the function
f  x   x  3 x  1 x  2
using:
x-intercepts:  3,0 ,  1,0 ,  2,0
y-intercept:  0, 6
 3, 1  2,  