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Transcript
Polynomials
Investigating Graphs and
Transformations
1/5/17
6-5
Finding Real Roots of
Polynomial Equations
Sometimes a polynomial equation has a factor
that appears more than once. This creates a
multiple root. In 3x5 + 18x4 + 27x3 = 0 has two
multiple roots, 0 and –3. For example, the root 0
is a factor three times because 3x3 = 0.
The multiplicity of root r is the number of times
that x – r is a factor of P(x). When a real root has
even multiplicity, the graph of y = P(x) touches the
x-axis but does not cross it. When a real root has
odd multiplicity greater than 1, the graph “bends”
as it crosses the x-axis.
Holt McDougal Algebra 2
6-5
Finding Real Roots of
Polynomial Equations
You cannot always determine the multiplicity of a
root from a graph. It is easiest to determine
multiplicity when the polynomial is in factored
form.
Holt McDougal Algebra 2
6-6
Fundamental Theorem of Algebra
You have learned several important properties
about real roots of polynomial equations.
You can use this information to write polynomial
function when given in zeros.
Holt McDougal Algebra 2
6-6
Fundamental Theorem of Algebra
Notice that the degree of the function in Example
1 is the same as the number of zeros. This is true
for all polynomial functions. However, all of the
zeros are not necessarily real zeros. Polynomials
functions, like quadratic functions, may have
complex zeros that are not real numbers.
Holt McDougal Algebra 2
6-7
Investigating Graphs of
Polynomial Functions
Polynomial functions are classified by their
degree. The graphs of polynomial functions are
classified by the degree of the polynomial.
Each graph, based on the degree, has a
distinctive shape and characteristics.
Holt McDougal Algebra 2
6-7
Investigating Graphs of
Polynomial Functions
End behavior is a description of the values of
the function as x approaches infinity (x
+∞)
or negative infinity (x
–∞). The degree and
leading coefficient of a polynomial function
determine its end behavior. It is helpful when
you are graphing a polynomial function to
know about the end behavior of the function.
Holt McDougal Algebra 2
6-7
Investigating Graphs of
Polynomial Functions
Holt McDougal Algebra 2
6-8
Transforming Polynomial Functions
You can perform the same transformations on
polynomial functions that you performed on
quadratic and linear functions.
Holt McDougal Algebra 2