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§1.3 Polynomial Functions

The student will learn about:

Polynomial function equations, graphs and roots.

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Introduction to PolynomialsA polynomial is an equation of the form

f (x) = an xn + an-1 x

n-1 + … + a1 x + a0

And graphs as a curve that wiggles back and forth across the x-axis.

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Definition Def: A polynomial function is a function of the form

f (x) = an xn + an-1 x

n-1 + … + a1 x + a0

For n a nonnegative integer called the degree of the polynomial.

The coefficients a0, a1, … , an are real

numbers with an ≠ 0.

The domain of a polynomial function is the set of real numbers.

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Graphs of a Polynomials

The shape of the graph of a polynomial function is connected to the degree and the sign of the leading coefficient an , and usually

wiggles back and forth across the x-axis.

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Graphs of a Polynomials Knowing the behavior of the ends of the graph helps us graph. What happens to y as x becomes very large or very small? That is, what happens at the tails?

Polynomial: Tail Behavior Chart

Leading coefficient positive

Leading coefficient negative

Degree even Both tails go up Both tails go down

Degree odd Left down, right up Left up, right down

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Graphs of a Polynomials

Polynomial: Tail Behavior Chart

Leading coefficient positive

Leading coefficient negative

Degree even Both tails go up Both tails go down

Degree odd Left down, right up Left up, right down

Remembering y = x 2 and y = x 3 may help!

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Graphs of a Polynomials The graph of a polynomial function of positive degree n can cross the x-axis at most n times.

An x-intercept is also called a zero or a root.

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Polynomial Root Approximation.

Theorem 1: If r is an x-intercept of the polynomial

P (x) = an xn + an - 1 x

n - 1 + an - 2 xn - 2 … + a1 x + a0

Let a n = the leading coefficient, and

Then |r| < (1 + | b | ) / | an | ) .

Let b = absolute value of the largest coefficient -

This gives us the maximum and minimum values for roots and helps in our search.

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Solving a Polynomial

Solutions may be found algebraically. Let f (x) = 0 and solve for x :

a. Factor – a bit of work that sometimes requires synthetic division (whatever that is).

b. Use the quadratic formula when you have second degree factors.

c. Use a graphing calculator and use the calc and zero buttons.

I love my calculator!

Remember you are responsible for both algebraic and calculator methods.

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Graphs of a Polynomials

The graph of a polynomial has origin (point, odd) symmetry, if all of the exponents are odd.

The graph of a polynomial has y-axis (even, vertical) symmetry if all of the exponents are even.

y = x 4 – 2x 2 - 1

y = x 3 – 2x

Note: both (x, y) and – x, y) are on the graph

Note: both (x, y) and (- x, - y) are on the graph

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Graphs of a Polynomials

A polynomial function is continuous with no holes or breaks.

The graph of a polynomial function of positive degree n can have at most (n – 1) turning points. These points are called relative maximum and relative minimum points.

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Exampley = x4 + 2 x3 – 4x2 - 8 x

I will factor using grouping

x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)

1. x-intercepts

Hence the x-intercepts are 0, 2, and – 2 as a root of multiplicity two.

I will factor using grouping

x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)

= (x 3 – 4x)(x + 2)

I will factor using grouping

x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)

= (x 3 – 4x)(x + 2) = x ( x 2 – 4)(x + 2)

= x (x – 2)(x + 2)(x + 2)

I will factor using grouping

x4 + 2x3 – 4x2 - 8 x = x 3 (x + 2) – 4x (x + 2)

= (x 3 – 4x)(x + 2) = x ( x 2 – 4)(x + 2)

Note that – 8 < r < 8. |r| < ( 1 + | b | ) / | an |

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Exampley = x4 + 2 x3 – 4x2 - 8 x

2. y-intercept

Allowing x = 0 in the original equation gives a y = 0 which means that the y-intercept is the origin,

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Exampley = x4 + 2 x3 – 4x2 - 8 x

3. Tail behavior.

The degree is even and the leading coefficient is positive so both ends go up.

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Exampley = x4 + 2 x3 – 4x2 - 8 x

3. Symmetry.

The function is neither even nor odd so it has no symmetry.

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Note: We will be finding the two relative minimum and the relative maximum with calculus although you already know how to do this with a graphing calculator!

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Polynomial Function Reviewf (x) = an x

n + an-1 xn-1 + … + a1 x + a0

The shape of the graph is connected to the degree and the leading coefficient.

Finding the x-intercepts is important.

The graph wiggles back and forth across the x-axis.

Finding the relative minimums and maximums will become important.

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Graphs of a Polynomials

Graphing polynomials is difficult and time consuming. Calculus will aide us greatly in determining the x-intercepts, (the y-intercept is always easy!), and the maximum and minimum points of a polynomial. Although this is helpful in graphing it is really more helpful in life as these characteristics have many applications.

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Summary.

• We had an introduction to polynomial functions and learned some of the properties of these functions.

• We had an introduction to rational functions.

• We learned about both the vertical and horizontal asymptotes associated with rational functions.

• We worked through an application that involved rational functions.

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ASSIGNMENT

§1.3; Page 13; 1 – 27 odd.