Polynomial Functions

1
Polynomial Functions

• NOTE
• A constant function f(x) c, where c ? is a
  polynomial function of degree 0
• A linear function f(x) mx b, where m ? 0 is a
  polynomial function of degree 1.
• A quadratic function f(x) ax2 bx c, where a
  ? 0 is a polynomial function of degree 2

2
Polynomial Functions

• Determine which of the following are polynomials.
  For those that are, state the degree
• f (x) 2x2 -5x 6
• g(x) 3vx - 5
• f(x) 2x2 / (6-3x)

3
End Behavior of Polynomial Functions
4
Leading Coefficient Test
Graph f(x) -x4 8x3 4x2 2 using a graphing
calculator with a viewing window of -8, 8 by
-10, 10. Does the graph show the end behavior
of the function ?
5
Example End Behavior

• Determine the end behavior of the following
• f(x) 3x6
• g(x) -5x4
• h(x) ½x 9
• S(x) -6 x 8

6
Example Leading Term
Determine the leading term of f(x)
(2x-1)2(x3)3(x-4)
Leading Term (2x)2 (x)3 (x) 4x6
7
Turning Point Theorem

• If f(x) is a polynomial function of degree n,
  then f(x) has at most n 1 turning points or
  local extrema.

• Ex 1 f(x) 3x2 2x 4
• Ex 2 f(x) 2x3 3x2 4x -2

8
Factor Theorem

• Let f(x) be a polynomial function. Then x c
  is a factor of f(x) if and only if f(c) 0.

Ex Given f(x) x2 3x 18, (x 6) is a
factor of f(x) if and only if f(-6) 0. (-6)2
3(-6) 18 0 Ex Given f(x) x2 3x 18,
(x - 3) is a factor of f(x) if and only if f(3)
0. (3)2 3(3) 18 0
9
Real Zeros of A Polynomial
To find the roots or zeros, factor the polynomial
and use the Zero-Factor Property
See bottom of Page 254 in text
10
Intermediate Value Theorem
Let f denote a polynomial function. If a lt b and
if f(a) and f(b) are of opposite sign, then there
is at least one zero of f between a and b. or
equivalently If P is a polynomial function and
P(a) and P(b) have opposite signs, then there
exists at least one value c between a and b such
that P(c) 0
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Example - Zeros

• Consider the polynomial in factored form.
  Determine the zeros of the polynomial using the
  zero-product property
• f(x) (3x-4)2(x1)3(x-3)

• Set each factor to zero and solve
• 3x 4 0 or x 1 0 or x 3 0
• x 4/3 or x -1 or x 3

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Multiplicity of Solutions

• If (x r)m is a factor of a polynomial function
  f(x) and (x r)m1 is not a factor of f(x), then
  r is called a zero of multiplicity m of f(x). In
  other words the solution x r occurs m times.
• Ex f(x) (x 2)3(x 1), then
• x 2 is a solution of multiplicity 3.
• x -1 is a solution of multiplicity one.

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Steps for Analyzing the Graph of a Polynomial

• Step 1 (a) Find the x-intercepts, if any, by
  solving the equation f(x) 0.
• (b) Find the y-intercept by finding f(0).
• Step 2 Determine whether the graph touches or
  crosses the x axis.
• Step 3 End Behavior find the power function
  that the graph of f(x)
• resembles for large values of x.
• Step 4 Determine the maximum number of turning
  points.
• Step 5 Find the intervals for which f(x) is
  positive or negative.
• Step 6 Plot the points in Steps 1 and 5 and the
  other information to
• connect them with a smooth, continuous curve.

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Example
a.) Find the x intercepts y-intercepts of
f(x). b.) Determine whether the graph crosses
or touches the x-axis at each
x-intercept. c.) Evaluate the end behavior of
the function. d.) Determine the maximum number
of turning points on the graph of f e.) Use the
x-intercepts and test numbers to find the
intervals on which the graph of f is above
the x-axis and the intervals on which the
graph is below the x-axis. f.) Put all the
information together to obtain the graph f(x)
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Graphing Polynomial Function
For f(x) (2x-1)2(x3)3(x-4)

• Find the x- and y-intercepts of the graph of f.
• Determine whether the graph crosses or touches
  the x-axis at each x-intercept.
• Find the power function that the graph of f
  resembles for large values of x. Determine the
  end behavior.
• Determine the maximum number of turning points on
  the graph of f.
• Use test points in each interval defined by the
  zeros to sketch a graph of f. Verify the graph
  using your calculator.

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Algebraic long division
Divide 2x³ 3x² - x 1 by x 2
x 2 is the divisor
2x³ 3x² - x 1 is the dividend
The quotient will be here.
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Remainder Theorem

• Let f(x) be a polynomial function. If f(x) is
  divided by x c, then the remainder is f(c).

Ex f(x) x3 4x2 2x 5 Divide by x 3 and
the remainder is 8. Evaluate f(3) - 8 Thus,
a factor which produces a remainder of zero
represents a solution to f(x).
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Finding Rational Zeros

• Example Find the rational zeros for f(x)
  x3 x2 10x 8
• p/q represents all possible zeros where p are all
  factors of the constant, and q are all factors of
  the first coefficient.
• p all the factors of 8 q
  all factors of 1
• 1, 2, 4, 8 1
• Here are all of the possible zeros for the
  function

p 1, 2, 4, 8 q 1

• Possible zeros are 1, 2, 4, 8
• So which one do you pick? Pick any.Find out
  which one is a zero by using synthetic division

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Example f(x) x3 x2 10x 8 possible
zeros 1, 2, 4, 8
Finding Rational Zeros

• Lets try 1

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Finding Rational Zeros

• Example Find the zeros of g(x) 6x3 4x2
  14x 4

Factors of p 1, 2, 4 Factors of q
1, 2, 3, 6
Possible zeros are 1, 2, 3, 4, 4/3,
1/2, 1/3, 1/6, 2/3
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Finding Rational Zeros
1 6 4 14 4 6
10 4 6 10 4 0 1
is a zero, and 6x2 10x 4 is the depressed
polynomial. Factor 6x2 10x 4 2(3x2 5x
2)
Try 1
2(3x 1)(x 2)
Find the zeros 2(3x 1)(x 2) 0
3x 1 0 x 2 0 x 1/3 x
2 The zeros are 1, 1/3, and 2
23
Finding Rational Zeros

• Example Find the rational zeros for f(x)
  x3 x2 10x 8
• p/q represents all possible zeros where p are all
  factors of the constant, and q are all factors of
  the first coefficient.
• p all the factors of 8 q
  all factors of 1
• 1, 2, 4, 8 1
• Here are all of the possible zeros for the
  function

p 1, 2, 4, 8 q 1

• Possible zeros are 1, 2, 4, 8
• So which one do you pick? Pick any.Find out
  which one is a zero by using synthetic division

24
Example f(x) x3 x2 10x 8 possible
zeros 1, 2, 4, 8
Finding Rational Zeros

• Lets try 1

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Finding Rational Zeros

• Example Find the zeros of g(x) 6x3 4x2
  14x 4

Factors of p 1, 2, 4 Factors of q
1, 2, 3, 6
Possible zeros are 1, 2, 3, 4, 4/3,
1/2, 1/3, 1/6, 2/3
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Finding Rational Zeros
1 6 4 14 4 6
10 4 6 10 4 0 1
is a zero, and 6x2 10x 4 is the depressed
polynomial. Factor 6x2 10x 4 2(3x2 5x
2)
Try 1
2(3x 1)(x 2)
Find the zeros 2(3x 1)(x 2) 0
3x 1 0 x 2 0 x 1/3 x
2 The zeros are 1, 1/3, and 2
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The Fundamental Theorem of Algebra

• Every complex polynomial function f(x) of degree
  n gt 1 has at least one complex zero.
• Because any real number is also a complex number,
  the theorem applies to polynomials with real
  coefficients as well.

The zeros may be real or complex... Counting
complex and repeated solutions, an nth degree
polynomial equation has exactly n solutions.
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Complete Factorization Theorem
Every complex polynomial function f(x) of degree
n gt 1 can be factored into n linear factors (not
necessarily distinct) of the form f(x) an(x -
r1)(x - r2).(x - rn) where an, r1, r2, , rn
are complex numbers (possibly real).
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Conjugate Zeros Theorem
If the polynomial P has real coefficients, and if
the complex number a bi is a zero of P, then
the complex conjugate a-bi is also a zero of P.
Linear Quadratic Factors Theorem
Every polynomial with real coefficients can be
factored into a product of linear and
irreducible quadratic factors with real
coefficients.