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Pre-Calculus Notes:
3.6, cont’d
Name: _________________________________
Helpful Hints for Synthetic Division…
How can I choose which of my rational zeros to try?
Descartes’ Rule of Signs:
Let f denote a polynomial function written in standard form (that’s highest  lowest power)
 The number of positive real zeros of f either equals the number of changes in the sign of
the nonzero coefficients of f(x)…or that number less an even integer.
 The number of negative real zeros of f either equals the number of changes in the sign of
the nonzero coefficients of f(-x)…or that number less an even integer.
(Note: if the sum of the coefficients is zero, “1” is a zero of the function!)
Example: Use the number of real zeros theorem and Descartes’ Rule of Signs to discuss the real zeros of f(x).
f ( x)  2 x 6  x 4  2 x 3  3 x  1
Number of Real Zeros: ____, (at most – that’s the degree), ____, _____, or _____ (we’re counting down by 2’s
because…
Number of Posi Real Zeros: Number of sign changes in f(x) = _____ or _____(we’re ALWAYS counting down by 2!)
Number of Negative Real Zeros: Number of sign changes in f(-x)
First, find f(-x):
Number of sign changes in f(x) = _____
Here’s another helpful hint!
Upper Bound Theorem
If p(x) is divided by x – c and there are no sign changes in the quotient or remainder, c is the upper bound (GO LOWER)
Lower Bound Theorem
If p(x) is divided by x - c and there are alternating sign changes in the quotient and remainder, c is the lower bound (GO HIGHER)
Let’s try using Descarte’s Rule of Signs and Upper Bound and Lower Bound Theorem on a problem:
f ( x )= x 4 - 3x 3 - 6 x 2 + 28x - 24
Maximum number of real zeros: _________ (but could also be _____ or _____)
Number of Positive Real Zeros (number of sign changes in f(x)): _________
Number of Negative Real Zeros (number of sign changes in f(-x)): _________
List possible rational zeros:
Test a number from the list. If it is a zero, decide if it is an upper or lower bound. (This isn’t required, but it can help you
decide other numbers to test!):
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Write in f(x) factored form, solve, and sketch the graph. Include y-intercept in your graph.
x 4  18 x 2  32 x  15  0
Now you try…Solve and sketch the equation in the real number system.
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c) Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
f)
Using part ‘d’ sketch the graph
a) Num of possible positive zeros:
Num of possible negative zeros:
b) possible rational zeros:
c)
d) write in factored form:
e) solutions:
f)
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Pre-Calc Review 3.2, 3.6
Polynomial Functions
Name:__________________________________
Date:________________ Period:___________
For each of the following a) Determine if the function is a polynomial (P) or not a polynomial (N) b) if it is a polynomial,
state the degree of the polynomial and the leading coefficient, then sketch its end behavior.
1. f  x   x5  2x 2  5x
2. f  x   2 x2 ( x 1)2 (3x  2)3
a)
a)
b)
b)
Sketch the graph of the following functions using transformations of power functions.
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3. g  x   2 x 2  3
4. f  x    x  2   3
Sketch and describe the end behavior of the following graphs.
5. f  x   3x2  2 x3  9
6.
End Behavior:
h  x   7 x 4  x 3  3x 2  4 x
End Behavior:
lim f ( x) =
lim f ( x) =
lim f ( x) =
lim f ( x) =
x ®¥
x ®¥
x ® -¥
x ® -¥
Find a polynomial whose zeros and degrees are given. (you may leave answer in factored form )
7. Zeros: -2, -1, 5; degree: 3
8. Zeros: -3, multiplicity of 2; 7, multiplicity of 2; degree 4
9. Zeros: -2, multiplicity of 1; 0, multiplicity of 3; 4, multiplicity of 2; degree 6
Use the Factor Theorem to determine whether x  c is a factor of f  x  (yes or no)
10.
f  x   2x4  8x3  x  10; x 1
11.
f  x   2 x3  x2  25x 12; x  3
4
For the following graphs fill out the information and then sketch a graph. Make sure to label your x and y axes.
12. f  x    x2  x  3 x  5
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
y-int-_________
Symmetry-_________
Interval above x-axis___________________
Interval below x-axis________________
End Behavior
Degree of polynomial _______
lim f ( x)  _________
x 
lim f ( x)  _________
x  
13. f  x    x  1
3
 x  2   x  3
2
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
x-int-___________ multiplicity-___________ cross or touches
y-int-_________
Symmetry-_________
Interval above x-axis___________________
Interval below x-axis________________
End Behavior
Degree of polynomial _______
lim f (x) = _________
x ®¥
lim f (x) = _________
x ® -¥
For each of the following functions:
a) Tell the maximum number of zeros that each polynomial may have.
b) List the potential rational zeros of each function
c) Then use Descartes rule of signs to determine how many possible positive and negative zeros each polynomial
function may have. You do not need to find the zeros.
14.
f  x   4 x 4  x3  x 2  2
15.
f  x   3x5  x3  4 x 14
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Use the Intermediate Value Theorem to show that each polynomial function has a zero in the given interval.
16. f  x   3x3  x2  25x 12;  3, 2
17. f  x   2 x3  5x  3; 1, 2
18. Solve and sketch the equation in the real number system. 4 x  4 x  11x  6 x  9  0
This means to do the following.
a) Use Descartes rule of sign to show how many possible positive and negative roots there are
b) Use Rational Zero Theorem to show what the possible rational zero’s are
c) Use synthetic or long division to test possible rational zero’s
d) Factor over the real number line. (x-a)(x-b)(x-c)=0
e) Solve over the real number line. (x = ??)
f) Using part ‘d’ sketch the graph
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a) Num of possible positive zeros:
Num of possible negative zeros:
3
2
f)
b) possible rational zeros:
c)
d) write in factored form:
e) solutions:
19. Find the maximum possible number of zeros, then use Descartes rule of signs to show how many possible positive and
4
3
2
negative roots there are: y = x  x  2 x  4 x  8  0
20. Find k such that f(x) = x4 + kx3 + 2 has the factor (x + 1)
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Use f ( x ) = x - 4 x - 7 x +10 to answer the following questions:
3
2
Use the remainder theorem and synthetic division to evaluate f(x) for x = 0. Is it a zero?
Use intermediate value theorem to show that f(x) has a zero on the interval [0, 2]
Factor f(x) completely given (x + 2) is a factor.
Sketch the graph of f(x) based on its zeros and end behavior. Include the y-intercept.
Circle the turning points on the graph above. Label them as local min / max.
Enrichment:
Use your calculator to find the ordered pairs of the turning points of f(x). (You may need to change your
window or zoom out). Label the ordered pairs on your graph (round to the nearest hundredth).
On what interval(s) is the graph increasing?
On what interval(s) is the graph decreasing?