Trig / Coll. Alg.
Name: ________________________
Chapter 3 – Polynomial Functions
3.1 Quadratic Functions (not on this test)
For each parabola, give the vertex, intercepts (x- and y-), axis of symmetry, and sketch the graph.
1. f ( x) = x 2 − 4 x − 5
2
2. f ( x) = −2 ( x + 4 ) + 8
3. Write the equation (in vertex form) of the parabola having vertex (3, 2) that contains the point (5, 4).
3.2b Zeros, Multiplicity and Graphing Polynomial Functions
Identify the left- and right-hand behavior of each function.
Review:
A)
f(x) = –x3 + 4x
B)
f(x) = x4 – 5x2 + 4
left:
left:
right:
right:
maximum number of turns:
maximum number of turns:
Determine the intervals over which the function is increasing, decreasing and/or constant:
C)
Increasing:
Decreasing:
Constant:
***Notice that the transformations we have worked with in the past for quadratic and absolute value
functions, remain the same for power functions.
Review:
Given f(x) is a power function:
a)
f(x) + a
results in a ________________ shift of ____ units
b)
g(x) = f(x + b)
results in a ________________ shift of ____ units
c)
g(x) = –f(x)
results in a ________________ over the __________
Features of Graphs of Polynomial Functions:
Graph is continuous – no breaks only smooth turns.
A graph of a polynomial function has at most ________ turns where n is the _____________ of the
polynomial.
A polynomial function of degree n has ______ zeros, although not all of the zeros must be
___________ _________________. Some zeros may be _____________ numbers.
I. Use your Graphing Calculator to graph:
f ( x) = x3 − 6 x 2 + 9 x
What are the x-intercepts of the graph?
These are also called the ____________ of the
function or the _________________ or
____________ of the equation (when f ( x) = 0 ).
Notice that the graph _____________ through the x-axis at ____ but only
____________ the x-axis at _____ .
Algebraically find the zeros of the function and compare these to your graph.
{Find the values of x for which f ( x) = 0 }
Multiplicity: _____________________________________________________________
The graph of a polynomial function will CROSS an x-intercept (zero) having _______ multiplicity, and
only TOUCH an x-intercept (zero) that has __________ multiplicity.
II. Find all the real zeros of the polynomial function, determine the multiplicity of each, and determine
whether the graph crosses or touches at each zero.
1. h(t ) = t 2 − 6t + 9
2. g ( x) = −2 x3 + 6 x 2 −
3. p(r ) = r 5 + r 3 − 6r
9
x
2
4.
a. Find a polynomial function that has the given zeros: 0, –2, 1
b. Find another polynomial function that has the same zeros.
III. Graph the polynomial function WITHOUT using your graphing calculator.
Steps:
Use the degree and leading coefficient to determine the general shape and end behavior of the
graph.
Determine the zeros of the polynomial and their multiplicities.
Plot at least one point between each zero and draw a continuous curve through the points.
5. f ( x) = x 4 − x 2
-10
6. f ( x) = −4 x3 + 4 x 2 + 15 x
10
10
5
5
-5
5
-10
10
-5
5
-5
-5
-
-
10
7. h( x) = x 2 ( x − 4 )
5
-10
-5
5
-5
-
10
10
3.3 – Synthetic & Long Division - Factoring and Zeros
Review: Find all the real zeros of the polynomial function and determine the multiplicity of each zero.
A.
f ( x) = x 4 − x 3 − 20 x 2
C.
h( x) =
D.
Find a polynomial of degree 2 that has a zero of –2.
1 2 5
3
x + x−
2
2
2
B.
g ( x) = x 2 + 10 x + 25
(Hint: factor out ½ )
(
)
I. Use synthetic division to divide 3x3 − 17 x 2 + 15 x − 25 by ( x − 5) . Give the quotient and
remainder.
Is ( x − 5) a factor of the polynomial?
(
)
2. Use synthetic division to divide x 6 − 4 x 4 + 3x 2 + 2 by ( x + 1) . Give the quotient and remainder.
Is ( x + 1) a factor of the polynomial?
(
)
If a polynomial f ( x) is divided by x − k ,
Remainder Theorem:
the remainder is f (k ) .
(
)
3. Determine the remainder without dividing: 3x3 + 8 x 2 + 5 x − 7 ÷ ( x + 2 ) .
***Note: synthetic division = synthetic substitution
Use synthetic substitution to evaluate:
4. a. f ( x) = 2 x3 − x 2 −10 x + 5 for x = 3
5. a. Find f (2) for f ( x) = x3 − 7 x + 6
b. Is ( x − 3) a factor of f ( x) ?
b. Is ( x − 2 ) a factor of f ( x) ?
c. Name a point on the graph of f ( x) .
c. Name a point on the graph of f ( x) .
Factor Theorem:
(
A polynomial f ( x) has a factor x − k
if and only if f (k ) =0.
)
6. Factor f ( x) completely and find the remaining zeros of f ( x ) given that:
f ( x) = 2 x 4 + 7 x3 − 4 x 2 − 27 x −18
and
( x − 2) & ( x + 3) are factors of
(
f ( x)
)
Synthetic division/substitution is easily used when the divisor is in the form x − k . If the divisor is
not in this form, however, long division can be used instead. The steps for long division of polynomials
are the same as the steps you used when you first learned how to do long division. It can be helpful to
insert 0’s for missing powers just as you would for synthetic division.
7. Use long division to divide 2 x 2 + 10 x + 12 by ( x + 3) .
8. Divide:
(x
3
) (
)
− 9 ÷ x2 +1
3.4a
Possible Rational Roots and Descartes’ Rule of Signs
I. Possible Rational Roots : If a polynomial function f ( x) has 1 or more rational roots, then the
root(s) will be in the form ±
p
where p = _______________________________________________
q
and q = _____________________________________________.
1.
List the possible rational roots of f ( x) = 2 x3 + 3x 2 − 8 x + 3 .
Use your graphing calculator to find which, if any, of the possible rational roots are zeros of the
function.
2.
List the possible rational roots of f ( x) = 2 x3 + 3x − 6 .
Use your graphing calculator to find which, if any, of the possible rational roots are zeros of the
function. What can you conclude about the zeros of this function?
Complex Zeros _____________ occur as ____________________ ____________.
In other words, if ______________ is a zero of f ( x) , then __________________ must also be
a zero of f ( x) .
3.
Find all of the zeros of f ( x) = x 4 − 3x3 + 6 x 2 + 2 x − 60 given that 1 + 3i is a zero. DO NOT
USE YOUR CALCULATOR!
II. Descartes’ Rule of Signs is a rule used to determine the possible number of “+” and “–“ zeros of a
polynomial function.
***Sign Variation: when the sign
on consecutive terms of the
Possible # of positive zeros: _______________________
polynomial changes from + to –, or
from – to +.
_______________________________________________
_______________________________________________
Possible # of negative zeros: _______________________________________________
_______________________________________________________________________
4. Use Descartes’ Rule of signs to find the possible number of positive and negative zeros of
f ( x ) = −3 x 7 + 4 x 4 + 3 x 2 − 2 x − 1
Positive:
Negative:
5. Use Descartes’ Rule of signs to find the possible number of positive and negative zeros of
g ( x) = 4 x3 − 3x 2 + 2 x − 1
Positive:
Negative:
6. Given f ( x) = 5 x3 − 2 x 2 − 10 x + 4 ,
a.) List the possible rational zeros.
b.) Use your graphing calculator to identify an actual rational zero, if possible.
c.) Use synthetic division to verify and find all the remaining zeros.
d.) Write the function in factored form.
3.5: Variations
If y varies directly with x, or y is directly proportional to x, then
proportionality or constant of variation. ( k ≠ 0 )
y = kx , where k is the constant of
If y varies inversely with x, or y is inversely proportional to x, then
k
x
y = , where k is the constant of
proportionality. ( k ≠ 0 )
When a variable quantity y is proportional to the product of two or more other variables, we say that y
varies jointly with these quantities. If y varies jointly with a and b, then y = kab .
Combinations of direct, joint, and/or inverse variation may occur. This is usually referred to as
combined variation.
1. The amount a spring will stretch, S, varies directly with the force (or weight), F, attached to the spring. If
a spring stretches 1.8 inches with 60 pounds attached, how far will it stretch with 90 pounds attached?
2. On planet X, an object falls 13 feet in 2 seconds. Knowing the distance it falls varies directly with the
square of the time of fall, how long does it take an object to fall 75 feet? Round your answer to three
decimal places.
3. The price per person of renting a bus varies inversely with the number of people renting the bus. It costs
$17 per person if 37 people rent the bus. How much will it cost per person if 78 people rent the bus?
4. The wattage rating of an appliance, W, varies jointly as the square of the current, I, and the
resistance, R. If the wattage is 2 watts when the current is 0.2 ampere and the resistance is 50
ohms, find the wattage when the current is 0.4 ampere and the resistance is 100 ohms.