Download - SECTION 3.1 POLYNOMIAL FUNCTIONS AND MODELS POLYNOMIAL FUNCTIONS AND MODELS

SECTION 3.1SECTION 3.1

POLYNOMIAL FUNCTIONS AND POLYNOMIAL FUNCTIONS AND MODELSMODELS

POLYNOMIAL FUNCTIONS


A polynomial is a function of the A polynomial is a function of the formform

f(x) = a f(x) = a n n x x nn + a + a n-1n-1 x x n-1n-1 + . . . + a + . . . + a11x x + a+ a00

where awhere ann, a , a n-1n-1, . . ., a, . . ., a11, a, a00 are real are real numbers and n is a nonnegative numbers and n is a nonnegative integer.integer.

The domain consists of all real The domain consists of all real numbers.numbers.



Which of the following are Which of the following are polynomial functions?polynomial functions?

43x - 2 f(x) x g(x)

1 - x2 - x

h(x)3

2

0 F(x)

8 G(x)



SEE TABLE 1SEE TABLE 1



The graph of every polynomial The graph of every polynomial function is smooth and function is smooth and continuous: no sharp corners and continuous: no sharp corners and no gaps or holes.no gaps or holes.



When a polynomial function is When a polynomial function is factored completely, it is easy to factored completely, it is easy to solve the equation f(x) = 0 and solve the equation f(x) = 0 and locate the x-intercepts of the locate the x-intercepts of the graph.graph.

Example:Example: f(x) = (x - 1)f(x) = (x - 1)22 (x + 3) = (x + 3) = 00

The zeros are 1 and - 3The zeros are 1 and - 3



If If ff is a polynomial function and is a polynomial function and rr is a real number for which is a real number for which f f ((r r ) ) = 0, then = 0, then rr is called a (real) zero is called a (real) zero of of f f , or root of , or root of ff..

If If rr is a (real) zero of is a (real) zero of f f , then, then

(a)(a) rr is an x-intercept of the is an x-intercept of the graph of graph of ff..

(b)(b) ((xx - - rr) is a factor of ) is a factor of ff..



If (x - r)If (x - r)mm is a factor of a is a factor of a polynomial f and (x - r)polynomial f and (x - r)m+1m+1 is not a is not a factor of f, then r is called a zero factor of f, then r is called a zero of multiplicity m of f.of multiplicity m of f.

Example: f(x) = (x - 1)Example: f(x) = (x - 1)22 (x + 3) = 0 (x + 3) = 0

1 is a zero of multiplicity 1 is a zero of multiplicity 2.2.



For the polynomial For the polynomial

f(x) = 5(x - 2)(x + 3)f(x) = 5(x - 2)(x + 3)22(x - 1/2)(x - 1/2)44

2 is a zero of multiplicity 12 is a zero of multiplicity 1

- 3 is a zero of multiplicity 2- 3 is a zero of multiplicity 2

1/2 is a zero of multiplicity 41/2 is a zero of multiplicity 4

INVESTIGATING THE ROLE OF MULTIPLICITY

INVESTIGATING THE ROLE OF MULTIPLICITY

For the polynomial f(x) = xFor the polynomial f(x) = x22(x - 2)(x - 2)

(a)(a) Find the x- and y-intercepts of the Find the x- and y-intercepts of the graph. graph.

(b)(b) Graph the polynomial on your Graph the polynomial on your calculator.calculator.

(c)(c) For each x-intercept, determineFor each x-intercept, determinewhether whether it is of odd or even it is of odd or even

multiplicity.multiplicity.

What happens at an x-intercept of What happens at an x-intercept of odd multiplicity vs. even odd multiplicity vs. even multiplicity?multiplicity?

EVEN MULTIPLICITYEVEN MULTIPLICITY

If r is of even multiplicity:If r is of even multiplicity:

The sign of f(x) does not The sign of f(x) does not change from one side to the change from one side to the other side of r.other side of r.

The graph The graph touchestouches the x-axis the x-axis at r.at r.

ODD MULTIPLICITYODD MULTIPLICITY

If r is of odd multiplicity:If r is of odd multiplicity:

The sign of f(x) changes from The sign of f(x) changes from one side to the other side of r.one side to the other side of r.

The graph The graph crossescrosses the x-axis the x-axis at r.at r.

TURNING POINTSTURNING POINTS

When the graph of a polynomial When the graph of a polynomial function changes from a function changes from a decreasing interval to an decreasing interval to an increasing interval (or vice versa), increasing interval (or vice versa), the point at the change is called a the point at the change is called a local minima (or local maxima). local minima (or local maxima). We call these points TURNING We call these points TURNING POINTS.POINTS.

EXAMPLEEXAMPLE

Look at the graph of f(x) = xLook at the graph of f(x) = x33 - - 2x2x22

How many turning points do How many turning points do you see?you see?

Now graph:Now graph:y = xy = x33, , y = xy = x33 - x, - x,

y = xy = x33 + 3x + 3x22 + 4 + 4

EXAMPLEEXAMPLE

Now graph:Now graph:y = xy = x44, , y = xy = x44 - - (4/3)x(4/3)x33, ,

y = xy = x44 - 2x - 2x22

How many turning points do How many turning points do you see on these graphs?you see on these graphs?

THEOREMTHEOREM

If f is a polynomial function of If f is a polynomial function of degree n, then f has at most n degree n, then f has at most n - 1 turning points.- 1 turning points.

In fact, the number of turning In fact, the number of turning points is either exactly n - 1or points is either exactly n - 1or less than this by a multiple of less than this by a multiple of 2.2.

GRAPH:GRAPH:

P(x ) = xP(x ) = x22 PP22(x) = (x) = xx33

PP11(x) = x(x) = x44 PP33(x) = x(x) = x55

When n (or the exponent) is even, the When n (or the exponent) is even, the graph on both ends goes to graph on both ends goes to

When n is odd, the graph goes in When n is odd, the graph goes in opposite directions on each end, one opposite directions on each end, one toward + toward + the other toward - the other toward - ..

EXAMPLE:EXAMPLE:

Determine the direction the Determine the direction the arms of the graph should arms of the graph should point. Then, confirm your point. Then, confirm your answer by graphing.answer by graphing.

f(x) = - 0.01xf(x) = - 0.01x 7 7

EXAMPLE:EXAMPLE:

Graph the functions below in the Graph the functions below in the same plane, first using [- 10,10] by same plane, first using [- 10,10] by [- 1000, 1000], then using [- 10, 10] [- 1000, 1000], then using [- 10, 10] by [- 10000, 10000]:by [- 10000, 10000]:

p(x) = x p(x) = x 55 - x - x 44 - 30x - 30x 33 + 80x + 3 + 80x + 3p(x) = x p(x) = x 55

The behavior of the graph of a The behavior of the graph of a polynomial as x gets large is polynomial as x gets large is similar to that of the graph of similar to that of the graph of the leading term.the leading term.

THEOREMTHEOREM

For large values of x, either For large values of x, either positive or negative, the graph of positive or negative, the graph of the polynomialthe polynomial

f(x) = a f(x) = a n n x x nn + a + a n-1n-1 x x n-1n-1 + . . . + a + . . . + a11x x + a+ a00

resembles the graph of the power resembles the graph of the power functionfunction

y = a y = a n n x x nn

EXAMPLEEXAMPLE

DO EXAMPLES 9 AND 10DO EXAMPLES 9 AND 10

CONCLUSION OF SECTION 3.1CONCLUSION OF SECTION 3.1

Download - SECTION 3.1 POLYNOMIAL FUNCTIONS AND MODELS POLYNOMIAL FUNCTIONS AND MODELS

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