Polynomial Functionsand Models Section 5.1

Polynomial FunctionsA polynomial function is a function of the form

Where are real numbers and is a nonnegative integer and .The degree of a polynomial is the largest power of .The leading coefficient is the coefficient of which is .

Is it a polynomial? If so, what is the degree?1. 2. 3. 4.

Example: Factors of a Polynomial• and are factors of • is a factor of • is a factor of • is a factor of

Zeros of a PolynomialIf is a function and is a real number for which , then is called a real zero of .As a result, the following are equivalent:• is a real zero of • is an -intercept of • is a solution to the equation • is a factor of

True or Falsea) 4 is a real zero of b) is a factor of c) 8 is an -intercept of d) If , then is a real zero of .

Exercise 42 – Page 342Find a polynomial of degree 3 with zeros: and 3.

True or FalseIf is a zero of then is also a zero of .

MultiplicityIf is a factor of the polynomial , then is called a zero of multiplicity of .Note: cannot be a factor of Example:

is a zero of multiplicity _______ of

Determine the real zeros and their multiplicity.1. Exercise 55 Page 342

2. Exercise 59 Page 342

𝑓 (𝑥 )=𝑘 (𝑥+6 )2 (𝑥+1 )3(𝑥+4) (𝑥−8 )21. List the real zeroes.2. Determine the multiplicity of each.3. At each real zero, determine if the graph touches or crosses.

The role of multiplicitySuppose is a real zero of a polynomial .If the multiplicity of is odd, then the graph of will cross the -axis at .If the multiplicity of is even, then the graph of will touch the -axis at .

True or FalseThe degree of the polynomial is even.

End BehaviorThe left side of the graph is down.The right side of the graph is up.

End BehaviorDescribe the end behavior for this graph.The left is ________ and the right is _________.

End BehaviorLeading coefficient is negative Leading coefficient is positive

Degree is odd Left upRight down Left downRight upDegree is even Left downRight down Left upRight up

Straight Line (odd degree)Leading coefficient is negative Leading coefficient is positive

Quadratic (even degree)Leading coefficient is negative Leading coefficient is positive

Is the degree even or odd?Is the leading coefficient or ?

Cubic ModelingSuppose we have a 10 inch by 20 inch piece of metal.A square will be cut out of each corner. The sides will be turned up to create a box.We want to determine the size of the square that should be cut out in order to maximize the volume of the box.

Find a formula for the volume of the box.

Trying various values for .Volume

1 234 2 352 3 378 4 336 5 250

Use the data points to fit a cubic that will model the volume of the box based on the amount that is cut out of each corner.Graph the model and determine the square that should be cut out to maximize the volume.