Section 2.3

Question 1

Determine whether the function is a polynomial function. If it is, identify the degree. f(x)= 9x^8 7x^5

Answer 1

In the definition, the exponent of each term is between 1 and n. Also note that if an intermediate power is missing from the function, the coefficient is understood to be 0. Look at the function f(x) = 9x8 7x5. Remember that when no exponent is written, it is understood to be 1. First, determine if all of the exponents in the function are nonnegative integers.
Notice that in the given function the exponents are 8 and 5. Therefore, all the exponents are nonnegative integers.
Now determine if all the coefficients are real numbers.
Notice that in the given function the coefficients are 9 and 7. Thus, the coefficients are real numbers.
Using the results from the previous step, the function f(x) = 9x8 7x5 is a polynomial. Therefore, f(x) = 9x 8 ÷ 7x 5 is a polynomial. The degree of the polynomial is 8 since the exponent with the greatest value is 8.

Question 2

How do you determine if a function is a polynomial?

Answer 2

The exponents have to be positive and the coefficients have to be real numbers.

Question 3

How do you determine the degree of a polynomial?

Answer 3

It is the highest exponent in the function.

Question 4

If h(x) is not a polynomial does it have a degree?

Answer 4

No.

Question 5

What does a polynomial function look like on a graph?

Answer 5

Two important features of the graphs of polynomial functions are that they are smooth and continuous. Smooth means that the graph contains only rounded curves with no sharp corners. Continuous means that the graph has no breaks and can be drawn without lifting your pencil from the rectangular coordinate system. Also, the end behavior of the graph must tend to rise or fall without bound.

Question 6

For n being odd, if the leading coefficient of a polynomial function is positive what does the graph look like?

Answer 6

The graph falls to the left and rises to the right.

Question 7

For n being odd, If the leading coefficient of a polynomial is negative what does the graph look like?

Answer 7

The graph rises to the left and falls to the right.

Question 8

For n being even, if the leading coefficient is positive what does the graph look like?

Answer 8

The graph rises to the left and the right.

Question 9

For n being even, if the leading coefficient is negative what does the graph look like?

Answer 9

The graph falls to the left and the right.

Question 10

What are zero’s of a polynomial function?

Answer 10

Zero’s are values of x for which f(x) is zero.

Question 11

How do you find the zeros of a polynomial function?

Answer 11

You set each factor equal to 0 and solve for x.

Question 12

What is the rule to find the multiplicity of a zero?

Answer 12

(x-r)^k with r being a zero and k being the degree of the factor. If the same factor x-r occurs k times , but not k+1 times, r is a zero with multiplicity k.

Question 13

What can be said of the multiplicity of a zero if it is even?

Answer 13

The graph touches the x axis at r.

Question 14

If r is a zero of odd multiplicity what can be said about the graph?

Answer 14

The graph crosses the x axis at r.

Question 15

Which number is smallest, 1, or -1?

Answer 15

-1.

Question 16

What is the multiplicity of 0?

Answer 16

1.

Question 17

If a factor repeats what is it’s multiplicity?

Answer 17

The sum of it’s degree’s.

Question 18

What does the intermediate value theorem state:

Answer 18

If f(a) and f(b) have opposite signs, then there is at least one value of c between a and b for which f(c) =0. The equation f(x)=0 has at least one real root between a and b.

Question 19

When do graphs tend to flatten out at zero’s?

Answer 19

When zero’s are greater than one.

Question 20

The graph of an even function in which f(-x) = f(x) is symetric with repect to what axis?

Answer 20

y axis.

Question 21

The graph of an odd function in which f(-x) = -f(x) is symetric with respect to where?

Answer 21

The origin.

Question 22

What is the maximum number of turning points on a graph rule?

Answer 22

n-1 where n equals the exponent of the first term.

Question 23

What does 3x^3 factor to?

Answer 23

3(x-0)^3