Download Algebra 2 Notes AII.7 Polynomials Part 2 Mrs. Grieser Name: Date:

Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser
Name: _______________________________________ Date: ________________ Block: _______
Zeros of a Polynomial Function
So far:
o If we are given a zero (or factor or solution) of a polynomial function, we can use
division to simplify the polynomial and find other zeros
What if we aren’t given a zero (or factor or solution)? How do we guess what might be a
zero (or factor or solution)?
The Rational Zero Theorem
If f(x) = anxn + an-1xn-1+…+a1x+a0 has integer coefficients, then every
rational zero of f has the form:
p
q
factor of constant term a 0
factor of leading coefficient a n
CAUTION: Not every possibility named by the Rational Zero Theorem will be a zero of
the polynomial function, but every rational zero of the polynomial will be in the list.
NOTE 1: Include positive and negative factors in the list of possibilities.
NOTE 2: It could be none of the possible zeros in the list are zeros of the function if the
function has no rational zeros.
Example 1: List all possible rational zeros of f(x) = 4x4 – x3 - 3x2 + 9x + 10.
o List factors of 10 (ao): _________________________________________________________
o List factors of 4 (an): ___________________________________________________________
o Create a list of possible zeros: __________________________________________________
Example 2: Find the zeros of f(x) = x3 - 8x2 + 11x + 20.
o List factors of 20 (ao): _________________________________________________________
o List factors of 1 (an): ___________________________________________________________
o Create a list of possible zeros: __________________________________________________
o Test them (how?) to see which might zeros _______________________________________
o If a zero has been found, divide or factor and find others:
You Try:
a) Find possible rational zeros for:
f(x) = x3 + 9x2 + 23x + 15
b) Find all real zeros for f(x) = x3 – 4x2 – 15x + 18
Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser Page 2
Fundamental Theorem of Algebra
Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least
one solution in the set of complex numbers.
Corollary: If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0
has exactly n solutions provided each solution repeated twice is counted as two
solutions, each solution repeated three times is counted as three solutions, and so
on.
Plain speak: A polynomial equation of degree n has n solutions (roots).
Some of the solutions may be imaginary
Repeated solutions are counted separately (multiplicity)
Note that if imaginary or irrational numbers are zeros of a function, then
their conjugates are also zeros.
Example 1: How many solutions does the equation x3 + 5x2 + 4x + 20 = 0 have?
The equation has 3 solutions. Explain______________________________
What are they?
o The rational zero theorem tells us possible rational solutions are: _________________
o Which possible rational zeros are solutions? ___________________________________
o Divide and find other solutions:
o The solutions are _________________
Example 2: Find the zeros of f(x) = x5 – 4x4 + 4x3 + 10x2 – 13x – 14
o The function has ___________ zeros
o Possible rational zeros ________________________________________________________
o Find the rational zeros and put function in factored form:
o Find complex zeros:
o List the zeros, including repeated zeros: _______________________________
You try: Find the zeros of f(x) = x5 – 2x4 + 8x2 - 13x + 6
Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser Page 3
Complex and Irrational Conjugates
Complex Conjugates Theorem
If f is a polynomial function with real coefficients, and a + bi is an imaginary zero of
f, then a – bi is also a zero of f.
Irrational Conjugates Theorem
Suppose f is a polynomial function with rational coefficients, and a and b are
rational numbers such that b is irrational. If a + b is a zero of f, then a - b is
also a zero of f.
Example: Write a polynomial function f of least degree that has rational coefficients, a
leading coefficient of 1, and 3 and 2 + 5 as zeros.
What are the zeros? _________________________________
Write f(x) in factored form: ___________________________
Expand to create polynomial:
Check by evaluating the polynomial at each zero:
You try: Write a polynomial function f of least degree that has rational coefficients, a
leading coefficient of 1, and zeros 3 and 3 – i.
Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser Page 4
Descartes’ Rule of Signs
To further help find zeros, mathematician René Descartes found the following:
Descartes Rule of Signs
Let f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0 be a polynomial function with real coefficients.
The number of positive real zeros of f is either equal to the number of sign changes of
f (x) or is less than that number by an even integer. If there is only one variation in
sign, there is exactly one positive real zero.
The number of negative real zeros of f is either equal to the number of sign changes of
f (-x) or is less than that number by an even integer. If f (-x) has only one variation in
sign, then f has exactly one negative real zero.
Use the rule of signs to help eliminate possibilities found with the rational zero theorem.
The maximum possible positive and negative zeros add to the degree of the polynomial.
Example: Use Descartes' Rule of Signs to determine the number of real zeros of:
f (x) = x5 – x4 + 3x3 + 9x2 – x + 5
1) Count sign changes in f(x):
There are 4 sign changes → 4 maximum possible positive real zeros
Maximum 4 positive zeros, but there could also be 2 or 0 positive real zeros
since some may be imaginary (count down by twos since theorem says even
integer zeros)
2) Count sign changes in f(-x)
Find f(-x) = (-x)5 – (-x)4 + 3(-x)3 + 9(-x)2 – (-x) + 5
= -x5 – x4 – 3x3 + 9x2 + x + 5
Count sign changes in f(-x):
There is 1 sign change → exactly one negative real zero
3) Conclusions: There are 4, 2, or 0 real positive zeros, and 1 negative real zero
You Try: Use Descartes’ rule of signs to determine number of real zeros or solutions:
a) Find number of real solutions:
4x7 + 3x6 + x5 + 2x4 – x3 + 9x2 + x + 1 = 0
b) Find number of real zeros:
f(x) = x5 + x4 + 4x3 + 3x2 + x + 1
c) Find the possible number of real solutions: 2x4 – x3 + 4x2 – 5x + 3 = 0
Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser Page 5
Graphing Polynomials, and Analyzing Graphs
So far: use end behavior and table of values
Now: put together what we have learned about finding zeros
More information about what happens in the middle of the graph (between ends)
Clarifications: what is the difference between zeros, x-intercepts, solutions, roots, and
factors??
The following statements are equivalent:
zeros
k is a zero of a function (x-value on graph where a function’s value is 0)
x-intercepts
If k is an x-intercept, it is the x-value on the graph where the function
crosses or touches the x-axis (very much like a zero)
k is a solution to f(x) = 0 (value of x that makes the equation true)
k is a root if it is a solution to f(x) = 0
(x – k) is a factor of f(x) if f(x) ÷ (x – k) has a remainder of 0
solutions
roots
factors
Generally speaking: functions have zeros, x-intercepts, and factors; equations have
solutions and roots.
Multiplicity of zeros is important. Zeros with EVEN multiplicity TOUCH the graph;
zeros with ODD multiplicity CROSS the graph.
1
( x 3)( x 2) 2
6
Example 1: Graph f ( x)
Fortunately the function is in factored form. If it
wasn’t, put it in factored form.
Graph the zeros. Zeros of the function are
__________________________
Note multiplicity of zeros – where does the graph
touch the x-axis, and where does it cross it?
___________________________
Plot some points between and beyond the
intercepts:
x
y
-2
-1
0
1
2
Determine end behavior:
o What is the sign of the leading coefficient and the degree of the
function?__________________
o As x → -∞, f(x) → ___________
o As x → +∞, f(x) → ___________
Draw graph by filling in curve through points plotted.
Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser Page 6
Turning Points
Location where function goes from increasing to decreasing
(or vice versa).
Turning points represent local maxima or local minima.
In the previous example, approximately where is the local
maximum? _______ Where is the local minimum? ________
A polynomial of degree n has at most n-1 turning points.
If a polynomial has n distinct zeros, then it has exactly n-1 turning points.
Determine intervals where functions increase or decrease by looking at turning
points.
o In the previous example, find the intervals where the function increases, and where it
decreases.
o Increases: ___________________________________
o Decreases: __________________________________
In the example at right, in what intervals does the
function increase? _____________________________
In the example at right, in what intervals does the
function decrease? _____________________________
Where is it neither increasing nor decreasing?
____________________
Note that finding the exact location of turning points
becomes much easier in calculus! For now, estimate, or use the graphing calculator to
help find exact locations of turning points.
You try:
a) Graph the function: f(x) = (x+1)2(x-1)(x-3)
zeros _______________
multiplicity: which zeros cross and which zeros touch xaxis? ________
plot some points in between and beyond zeros:
End behavior:
o
As x → -∞, f(x) → ___________
o
As x → +∞, f(x) → ___________
Draw graph
Algebra 2 Notes AII.7 Polynomials Part 2
Mrs. Grieser Page 7
b) Estimate the coordinates of each turning point and state whether the
turning point is a local maximum or local minimum. Estimate all real
zeros. Determine the least degree of the polynomial. Find the intervals
the function increases and then find the intervals it decreases.
Writing Polynomials from a Graph
Write the cubic function shown in the graph at right.
o Identify the zeros ________________________
o Write the function in factored form:

f(x) = _________________________________
o Find the value of a by substituting in a point that is not a
zero:
o Expand to polynomial form
You try:
Write a polynomial function whose graph passes through the given points:
(-1, 0), (0, -12), (2, 0), (3, 0)