Math 3 Name ______
3-2 Zeroes & Factors of Polynomials – Part 2
· I understand the relationship between standard and factored forms of polynomials
Equivalent Expressions for Polynomial Functions
Without using a calculator, answer the following questions about the function
1.) What is the factored form of?
2.) What are the zeros of?
3.) Sketch a graph of. Label the maximum/minimum and y-intercept.
Consider the function. How many of the above questions can you answer about ?
4. The function can be written as
Find the zeros of. Sketch a possible graph ofbelow.
Zeros:
Producing Useful Equivalent Expressions
Expressing cubic, quartic, and higher degree polynomials in factored form is a challenging problem. For example, the function
can be written as
Could you have found the linear factors on your own? This problem interested mathematicians for many centuries, and they developed a variety of special techniques for finding factors of polynomials with integer coefficients. With the development of modern computational tools, those formal techniques are now less important than they once were. However, both modern and ancient methods rely on the relationship between factors and zeros that you know well for quadratic polynomials.
5. The graph to the right is of the cubic polynomial
a. Use the graph to find the zeros of .
b. Use the zeroes to write in factored form.
c. Expand your answer to part (b) and show that your form is
equivalent to the above given equation.
6. Following the strategy you developed in problem (5), write the below polynomials in factored form.
You will need to graph the equations on your calculator.
a. b.
Sketch of Graph Sketch of Graph
Factored Form Factored Form
c. d.
Sketch of Graph Sketch of Graph
Factored Form Factored Form
e. Expand your answer to part (6a). Do you get the same equation as that for ?
f. Expand your answer to part (6c). Do you get the same equation as that for ?
g. Look at your factored form for parts (c). What is the leading coefficient of the original function? Is that the leading coefficient of ? If not, adjust your factored form by adding a common factor so that, when expanded, it will have the correct leading coefficient. Do the expansion again (you should NOT have to redo all the work . . .) to check if your equation now matches
h. Expand your answer to part (6d). Does your answer match? Will adding a leading coefficient make your answer match? How is the graph of different than the graph of the other functions? This difference must have some meaning . . .
STOP: Write a short summary of what you have learned in problems 5 and 6. Refer to your learning goal to write your summary!! You may not have all the answers at this point, but write what you notice!
To write the equation of a polynomial is factored form . . . special cases are. . . .
Your work thus far has showed you that using the zeroes of a polynomial function helps in writing a rule for
the function as a product of linear factors. But that rule might need some adjustment to make it match the
function exactly. For each of the following functions:
· Use the information about zeros to write a preliminary factored form of the rule
· Use your calculator to compare tables and graphs produced by the factored form and standard form rules.
· Adjust the factored form in a way that makes it equivalent to the standard form
7a. has zeroes of
7b. has zeroes of
7c. has zeroes of
STOP: Write a short summary of what you have learned in problem 7. Refer to your learning goal to write your summary!!
8. An important characteristic that is used to describe a polynomial function is the number of zeroes. For example, the polynomial function is said to have a zero of multiplicity 3 atand a zero of multiplicity 2 at .
The shape of the graph of a polynomial function near a zero provides information about the possible
multiplicity of the zero.
a. Graph the polynomial function for different positive integer values of n. Examine
the behavior of each graph near . What appears to be true about the behavior of a polynomial function near a zero of multiplicity n?
b. Test your conjecture in part (a) by considering the zeroes of the polynomial function
. Revise your answer to part (a) if necessary.
9. Examine the graph of at the right. The scale on both axes is 1. Find an equation for if . This graph has been stretched/compressed, so it should have a k value!!
10. Find the equation of the below graph:
10. Find the equation of the below graph given that and the non-integer zero of is .
Other than k, your final answer should be all integers!
SUMMARY: What is the relationship between standard form and factored form of a polynomial?
Your answer should have several parts – summarizing what we learned in this investigation cannot be done in one or two phrases!!