Math 3
3-2 Zeroes & Factors of Polynomials – Part 2
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Name ________________________
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 g x   x 2  2 x  3
1.) What is the factored form of g x ?
2.) What are the zeros of g x ?
3.) Sketch a graph of g x . Label the maximum/minimum and y-intercept.
Consider the function f  x   2 x4  5x3  26 x2  41x  60 . How many of the above questions can you
answer about f x  ?
4. The function f x  can be written as f  x    x  3 x  4 x  1 2 x  5
Find the zeros of f  x  . Sketch a possible graph of f  x  below.
Zeros:
Producing Useful Equivalent Expressions
Expressing cubic, quartic, and higher degree polynomials in factored form is a challenging problem. For
example, the function
f  x   x5 15x4  85x3  225x2  274 x 120
can be written as
f  x    x  1 x  2 x  3 x  4 x  5
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 c  x  .
b.
Use the zeroes to write c  x  in factored form.
c.
Expand your answer to part (b) and show that your form is
equivalent to the above given equation.
6.
a.
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.
f  x   x3  9 x2  11x  21
b. g  x   x4  4x3  52 x2 112 x  384
Sketch of Graph
Factored Form
f  x 
c.
h  x   3x3  21x 18
Sketch of Graph
Factored Form
g  x 
d. k  t   2t 4  4t 3  22t 2  24t  72
Sketch of Graph
Sketch of Graph
Factored Form
Factored Form
h  x 
k t  
e. Expand your answer to part (6a). Do you get the same equation as that for f  x  ?
f. Expand your answer to part (6c). Do you get the same equation as that for h  x  ?
g. Look at your factored form for parts (c). What is the leading coefficient of the original function? Is that
the leading coefficient of h  x  ? 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  x 
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 k  t  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:
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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.
2 1
f  x   6x3  5x2  3x  2 has zeroes of x   , , and 1
3 2
7b.
2
f  x   5x3 18x2  7 x  6 has zeroes of x   , 3, and 1
5
7c.
2
7
f  x   6 x3  17 x2  14 x has zeroes of x  0, , and 
3
2
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
3
2
example, the polynomial function f  x    x  1  x  2    x  1 x  1 x  1 x  2  x  2  is said to
have a zero of multiplicity 3 at x  1 and a zero of multiplicity 2 at x  2 .
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 f  x    x  3 for different positive integer values of n. Examine
n
the behavior of each graph near x  3 . 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
2
3
g  x    x  3  x  1  x  1 . Revise your answer to part (a) if necessary.
9.
Examine the graph of y  h  x  at the right. The scale on both axes is 1. Find an equation for h  x 
if h  1  3 . 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
1
g (1)  24 and the non-integer zero of g  x  is x  .
2
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!!