1. No category

Theory of Equations
Worksheet #1
A POLYNOMIAL FUNCTION
A function that contains only the operations of addition, subtraction, and multiplication
(including repeated multiplication) is a polynomial function.
When simplified, a polynomial function is usually written as:
f(x) = anxn + an-1xn-1 + an-2xn-2 + ... + a1x + a0,
where n is a positive integer (called the degree of the polynomial) and ak is any real
number for k = 0, 1, ..., n.
We want to examine the graph of a polynomial function as more and more factors are
multiplied together. Use your calculator and sketch a quick graph of each of the
following functions. Follow the sequence given. Use a reasonable window, set Yscl=0,
and mark the intercepts on the x-axis in the graphs below.
y1 = (x - 1)
y2 = (x - 1)(x + 2)
1
y3 = (x - 1)(x + 2)(x - 3)
y5 = (x - 1)(x + 2)(x - 3)(x + 4)(x - 5)
y4 = (x - 1)(x + 2)(x - 3)(x + 4)
y6 = (x - 1)(x + 2)(x - 3)(x + 4)(x - 5)(x + 6)
Write a few sentences describing what you observed about the sequence of functions
you just graphed.
What is happening to the degree of the polynomial?
How does the degree affect the graph?
2
Theory of Equations
Worksheet #2
We want to examine the role of the exponent on each factor and its effect on the graph
of the polynomial. Using your calculator, make a quick sketch of the graph of each of
the following functions. Use a meaningful window, and mark the x-intercepts with their
values.
y1 = (x + 2)(x - 1)(x - 3)
y2 = (x + 2)2(x - 1)(x - 3)
y3 = (x + 2)2(x - 1)2(x - 3)
y4 = (x + 2)2(x - 1)(x - 3)3
y5 = (x + 2)3(x - 1)(x - 3)4
y6 = (x + 2)5(x - 1)3(x - 3)2
3
Under what circumstances did you find a "pass-through" point? A "bounce" point?
Explain (clearly) the relationship between the exponents on each of the factors in the
polynomial functions and the behavior of the graphs at x = -2, 1, and 3.
Determine the degree of each polynomial on the other side. Recall what you wrote on
worksheet #1 about the degree and its effect on the graph. Are the graphs on this sheet
consistent with that information? Why or why not?
Under what circumstances do graphs cross the x-axis?
1.
Give a first degree equation that crosses the x-axis.
2.
Give a first degree equation that does not cross the x-axis.
3.
Give a second degree equation that crosses the x-axis two times.
4.
Give a second degree equation that touches the x-axis at one and only one point.
5.
Give a second degree equation that does not touch the x-axis at all.
4
6.
7.
8.
Now try this same idea for third degree equations:
Write a third degree equation that intersects the x-axis
a.
three times:
b.
twice:
c.
once:
d.
not at all:
Repeat this experiment using fourth degree equations:
Write a fourth degree equation that intersects the x-axis
a.
four times
b.
three times
c.
twice:
d.
once:
e.
not at all:
Try to generalize the above observations. Under what conditions will a
polynomial intersect the x-axis a given number of times. (You may want to
consider even degree and odd degree polynomials separately.)
5
Theory of Equations
Worksheet #3
Sketch the graph of each of the following equations. You may need to factor first.
1.
y  2 x 3  4 x 2  30 x
2.
y  x3  5 x 2  9 x  45
3.
y  4 x 4  73 x 2  144
4.
y  x 4  20 x 2  96
5.
y  x6  8x4  9 x2
6.
y  ( x  3)3 (2 x  5) 2 ( x 2  16)
6
Theory of Equations
Worksheet #4
Write an equation of a polynomial function for each set of given conditions. If the
situation is impossible, state this. In many cases, there are an infinite number of
possibilities. You may leave your answers in factored form.
1.
2.
3.
Linear equation (first degree)
a.
crosses the x-axis at x = 3
b.
bounces off the x-axis at x = -2
c.
does not touch the x-axis
Quadratic equation
a.
passes through the x-axis at x = -2 and x = 6
b.
bounces off the x-axis at x = -4
c.
does not touch the x-axis
Cubic equation (3rd degree)
a.
crosses the x-axis at x = -1, -4, and 3
b.
bounces off the x-axis at x = 2 and passes through at x = -3
c.
passes through x = 4 with no other zeroes
d.
does not touch the x-axis
7
4.
5.
Quartic equation (4th degree)
a.
crosses the x - axis at x = -2, 0, 3, and 7
b.
bounces off the x-axis at x = -2 and x = 3
c.
bounces off the x-axis at x = 1 and passes through at x = -3 and 4
d.
passes through x = -1 and x = 5 with no other zeroes
e.
does not touch the x-axis
Assorted...
a.
has a bounce point at x = 2, a cubic root at x = 0, and passes through at
x = -1.
b.
has bounce points at x = -3, 1, and 4 and y 0 for all x
c.
cubic, a linear root at x = 2, with no other points of intersection with the xaxis
d.
fourth degree, passes through linearly at x = -1 and x = 4, with no other
points of intersection with the x-axis
e.
cubic, with f(3)=0, f(1)=0 and positive only when x>3
8
Theory of Equations
Worksheet #5
Leading coefficients give you information about whether the right-hand side of the graph
points up or down, and how high or low the peaks and valleys of the graph are.
Sketch the following graphs:
1.
y  ( x  2)( x  3)
2.
y   ( x  2)( x  3)
What is the difference between the first and second equations?
What is the difference between their graphs?
3.
y  (2  x)(3  x )(5  x )
4.
y  ( 2  x )(3  x )(5  x)
What is the difference between the third and fourth equations?
What is the difference between their graphs?
9
5.
y  ( x  1)( x  3)
6.
y  4( x  1)( x  3)
What is the difference between the fifth and sixth equations?
What is the difference between their graphs?
7.
y  ( x  2)( x  1)
8.
y  3( x  2)( x  1)
What is the difference between the seventh and eighth equations?
What is the difference between their graphs?
Make some general statements about the effect the leading coefficient has on the
graph.
10
Theory of Equations
Worksheet #6
You can find the leading coefficient for a graph where a point other than one of the xintercepts can be determined.
Example:
A cubic equation has x-intercepts at 2, 4 and –3 and also contains the
point (1,24)
Write the general equation, using a variable (“a” is most common) for the leading
coefficient.
Here, y = a (x-2)(x-4)(x+3). Since the point (1,24) is on the graph it must “work” in the
equation, so substitute and solve for a.
24 = a (1-2)(1-4)(1+3)
24 = a (-1)(-3)(4)
24 = a (12)
2 =a
So the equation is y = 2(x-2)(x-4)(x+3)
Sketch the graph and find the equation described by each.
1.
quadratic, zeros at 4 and –5, contains the point (1, -54)
2.
quartic, f(0)=f(-1)=0, tangent to the x-axis at 3, f(4)=-40
3.
cubic, double root at 2, single root at –1/2, contains the point (-2,24)
11
Theory of Equations
Worksheet #7
Write the equation of the polynomial function for each set of given conditions.
1.
linear, containing (2, -5), with 4 as a zero
2.
linear, with -10 as a zero, and 8 as the y-intercept
3.
quadratic, containing P(5, 10), with -3 and 9 as zeroes
4.
quadratic, containing P(-4, 3), with 6 as its only zero
5.
quadratic, containing P(-7, 10), with 4 i and - 4i as its zeroes
6.
cubic, with zeroes -7, 2, and 9, with y-intercept 12
7.
cubic, zeroes at 2, 3 i , and -3 i , containing (-1, -15)
8.
quartic, a bounce point at 2, other zeroes at -3 and 0, containing (1, -16)
12
9.
Write cubic equations for the following:
(2,2)
-5
6
4
-4
2
10.
Write the quintic equations for the following.
(2,18)
5
11.
For the first graph in the problem (10), what other equations are possible if we do
not assume the polynomial is a quintic?
13
Theory of Equations
Worksheet #8
Review of Long Division:
Example:
10
x3
4
3
2
x  6 x  4 x  11x  2
x3  3x 2  5 x  4 
x 4  6x 3  4x2  11x  2

x3
x3
 x 4  3x 3 
 3x3  4x2
 3x3  9x2 
 5x 2  11x
 5x2  15x
 4x  2
 4x  12
 10
Find the quotient and the remainder for each problem.
1.
2 x3  5 x 2  8 x  5
x2
2.
6x2  7 x  3
2x  3
14
Theory of Equations
Worksheet #9
Review of Synthetic Division:
Example:
3 1
-6
3
4
-9
11
-15
2
-12
1
-3
-5
-4
-10
x  6x  4x  11x  2

x3
4
3
2
Then rewrite as: x3  3x 2  5 x  4 
10
(or use Remainder –10)
x 3
Find the quotient and the remainder for each problem.
1.
2 x 2  x  16
x 3
3.
x3  9 x 2  27 x  28
x3
2.
5x2  4 x  2
x5
4.
x2  4x  4
x2
15
5.
2 x3  2 x  3
x 1
6.
x 4  4 x3  10 x 2  12 x  9
x2  2 x  3
7.
x4  4
x2  2x  2
8.
x5  3 x 4  7 x3  11x 2  11x  13
x2  x  2
9.
x 6  4 x5  8 x 4  2 x3  20 x 2  17 x  30
x3  2 x 2  3
10.
4 x 7  3 x 6  12 x5  5 x 4  30 x 2  7 x  40
x 4  2 x3  2 x  5
16
Theory of Equations
Worksheet #10
Given: f ( x)  3 x 2  5 x  2
Find the remainder when f ( x ) is divided by x  2 .
Find the remainder when f ( x ) is divided by x  3
Evaluate f (2) and f ( 3) .
What is the relationship between the divisions and evaluations above?
Write a generalization about the relationship between the remainder when dividing f ( x ) by
x  a and the value of f ( a ) .
17
The Remainder Theorem
When a polynomial
is divided by
, the remainder is
.
This can be used to find remainders without dividing, and can also be used to evaluate
polynomial functions.
This is also sometimes called synthetic substitution.
1.
Given: f ( x)  4 x 4  3x3  2 x 2  8
Find:
a.
f ( 1)
b.
f (1)
c.
f (2)
2.
Given f ( x)  5 x30  2 x 25  7 x8  9 x 2  3x  2
Find the remainder when f ( x ) is divided by:
a.
x 1
b.
x 1
18
Theory of Equations
Worksheet #11
The Factor Theorem
For a polynomial
,
is a factor if and only if
Given P( x)  x 4  4 x3  7 x 2  22 x  24 , tell if each of the following is a factor of P( x)
1.
x3
2.
x2
3.
x 1
4.
x2
5.
x 1
6.
x4
19
7.
Given that x  3 is a factor of x 3  6 x 2  11x  6 , find the other factors.
8.
Given that x  1 is a factor of x 3  3 x 2  6 x  8 , find the other factors.
9.
Given that 3 is a root of x 3  6 x 2  x  30 , what is one of the factors of x 3  6 x 2  x  30 ?
What are the other factors of x 3  6 x 2  x  30 ?
What are the other roots of x 3  6 x 2  x  30 ?
10.
Given that 6 is a root of 6 x 3  37 x 2  4 x  12 , what are the other roots?
20
Theory of Equations
Worksheet #12
With the given polynomial and root(s) find the remaining roots:
1. 2 x 3  5 x 2  4 x  3  0 ; root x  3
2. 6 x3  11x 2  4 x  4  0 ; root x  2
3. 2 x 4  9 x3  2 x 2  9 x  4  0 ; roots x  1, x  1
4. 4 x 4  4 x3  25 x 2  x  6  0 ; roots x  2, x  3
21
5. x 4  3 x3  3 x 2  3 x  4  0 ; roots x   i
6. x 4  2 x 3  x 2  4  0 ; roots x  1, x  2
7. x 4  3x 3  19 x 2  87 x  90  0 ; double root of x  3
22
Theory of Equations
Worksheet #13
If you are given one of the roots of the equation 2 x 3  7 x 2  2 x  3  0 , finding the others is not
difficult. However, if you are not given one of the roots, you need to find a way to find it. To do
this you can use the Rational Root Theorem.
The Rational Root Theorem
If P(x) is a polynomial of degree n with integral coefficients and a nonzero constant term:
where
the rational roots of P(x) must be of the form
, where p is a factor of
and q is a factor of
.
In the equation from above, 2 x 3  7 x 2  2 x  3  0 , what are the values of a0 and an ?
Using the idea that p must be a factor of a0 , what are the possible values of p?
Using the idea that q must be a factor of an , what are the possible values of q?
Since the rational roots must be of the form
p
, what are the possible rational roots of this
q
equation?
Remember that if a value is a root of the equation, the remainder when dividing must be 0. Find
one of your possible rational roots that is a root of the equation.
23
You can now use the “depressed equation” (the remaining factor) to find the remaining roots.
What are the remaining roots of this equation?
Use this idea to solve the following:
1.
x3  3x 2  10 x  24  0
2.
6 x3  11x 2  3x  2  0
3.
2 x 4  5 x3  11x 2  20 x  12  0 (Note: how many roots do you need to find?)
24
Theory of Equations
Worksheet #14
Use factoring, the rational root theorem and synthetic division, or graphing to find the roots of
each equation. List all roots – if you have a double root, list it twice.
1. x3  x 2  x  1  0
2. x3  2 x 2  x  2  0
3. x 4  10 x 2  9  0
4. 2 x 3  9 x 2  3x  4  0
5. 3x3  4 x 2  5 x  2  0
6. 3x 4  2 x3  9 x 2  12 x  4  0
25
7. x 4  2 x 3  2 x 2  6 x  3  0
8. 3x 3  x 2  36 x  12  0
9. 2 x3  7 x  2  0
10. 2 x 4  x3  7 x 2  x  2  0
26
Theory of Equations
Worksheet #15
Using the problems from Worksheet #14, find the sum of the all of the roots of the
equation and the product of all of the roots of the equation. Leave your answers in
fractional form.
Equation
List ALL Roots
Sum of roots
Product of roots
x  x  x 1  0
3
2
x3  2 x 2  x  2  0
x 4  10 x 2  9  0
2 x3  9 x 2  3x  4  0
3x3  4 x 2  5 x  2  0
3x 4  2 x 3  9 x 2  12 x  4  0
x 4  2 x3  2 x 2  6 x  3  0
3x 3  x 2  36 x  12  0
2 x3  7 x  2  0
2 x 4  x3  7 x 2  x  2  0
What do you notice about the relationship between the sum of the roots and the
equation? Between the product of the roots and the equation? (Hint: look at the
coefficients of the original equation)
Find the sum and product of the roots of 8 x 4  38 x3  9 x 2  108 x  27  0
27
Theory of Equations
Worksheet #16
1.
Find a cubic equation with integral coefficients and roots of 1  2 and 4.
2.
Find a quartic equation with integral coefficients and roots of 3  2i and  3
3.
The equation x3  2 x 2  ax  b  0 has a root of 2  i . Find the other two roots.
4.
Find the values of a and b in problem #3.
5.
The equation x3  ax  b  0 has a root of 2  3 . Find the values of a and b.
28
6.
Find the values of a and b if the equation x3  ax 2  bx  10  0 has a root of
7.
The equation x3  ax  16  0 has double root. Find the roots of the equation.
8.
Find the value of a in the equation in #7.
9.
The equation 2 x3  x 2  32 x  a  0 has roots that are opposite each other. Find the value
of a.
10.
What are the roots of the equation in #9?
11.
A cubic equation has no quadratic term. If one root is 5  3 , what are the other roots?
12.
A quartic equation has integral coefficients, no cubic term and no constant term. If one
of the roots is 4  2i , what are the other roots?
5
29
Theory of Equations
Worksheet #17
Find the roots of x3  3 x 2  4 x  12  0
What are the reciprocals of the roots of the equation?
Write an equation whose roots are the reciprocals of the roots of x3  3x 2  4 x  12  0
What is the relationship in the coefficients between the original equation and the equation with
roots that are the reciprocals of the roots of the original equation?
Going back to the original equation, double each of the original roots and write a new equation
whose roots are double those of the original equation.
Now triple each of the original roots and write a new equation whose roots are triple those of the
original equation.
What is the relationship between the coefficients of the original equation and the coefficients of
the new equation when the roots are doubled or tripled?
(Hint: make sure all missing terms are included and your final answer is simplified if possible)
30
1.
Write an equation with roots that are twice those of 4 x 3  2 x 2  7 x  2  0
2.
Write an equation whose roots are the opposite of those of 3x 4  9 x3  7 x 2  5  0
3.
Write an equation whose roots are reciprocals of those of x 4  9 x3  7 x 2  5 x  2  0
4.
Write an equation whose roots are triple those of x3  2 x  3  0
5.
Write an equation whose roots are double those of 4 x3  x 2  1  0
6.
Write an equation whose roots are reciprocals of those of 3x 3  5 x  7  0
7.
Write an equation whose roots are opposite those of 2 x 4  7 x 2  3x  11  0
8.
The roots of the equation 5 x3  3 x 2  2 x  4  0 are tripled. Then the opposite of each
root is taken. What is the equation whose roots are the reciprocals of those new roots?
31
Theory of Equations
Worksheet #18
Review
1.
2.
Sketch the graph of each. You do not need to use a scale on the y-axis.
a.
f ( x)  ( x  3)( x  7)(2 x  5)
b.
f ( x)  ( x  3) 2 ( x  6)
c.
f ( x)  x 3 (2 x  7) 2 ( x  3)
d.
f ( x)  ( x 2  9)( x 2  8)
Write the equation for each graph. Include the leading coefficient.
a.
y-intercept at –2
b.
y-intercept at 1
32
c.
y-intercept at –2
d.
y-intercept at –3
3.
Write an equation for a graph with x-intercepts at –4 and 6 and y-intercept 4.
4.
Write a quadratic function with f (2)  0 , f (6)  0 and f (0)  24 .
5.
Write a cubic funtion with f (1)  f (4)  f (2)  0 and f (1)  6 .
6.
Write a quartic function whose graph is tangent to the x-axis at –2 and 3, and which has a
y-intercept of –4.
33
7.
Write a quintic function with zeros at 0, 1, 3 and 2i , and whose graph contains the point
(2,12) .
Solve each equation :
8.
x 4  21x 2  100
9.
x3  4 x 2  x  4  0
10.
x3  6 x 2  11x  6  0
11.
x 4  81  0
12.
4 x3  3x  1  0
13.
5 x 3  30 x 2  45 x  0
14.
6 x3  25 x 2  23 x  6  0
15.
x3  8
34
16.
x 4  3x 2  2  0
17.
x 4  2 x 3  8 x  16  0
18.
9 x3  23x 2  x  5  0
19.
x3  12 x 2  21x  10  0
20.
x 4  x3  x  1  0
21.
x3  4 x 2  7 x  6  0
22.
What is the remainder when 3x 3  5 x 2  5 x  3 is divided by x 2  2 x  3 ?
23.
If two of the solutions of x 4  4 x3  27 x 2  38 x  16  0 are 2 and 1, find the other
solutions.
24.
What is the sum of the solutions of 6 x 4  29 x3  40 x 2  12  0 ? What is the product of
the solutions ?
35
25.
Find the value of k if P (3)  1 when P ( x)  2 x 4  3x 2  kx  20
26.
The equation x5  12 x3  16 x 2  4 x  16  0 has roots of 1  3 and 1  i . What are the
other roots ?
27.
Two of the roots of 3x 3  17 x 2  ax  b  0 are 2 and 4. Find a, b, and the third root.
28.
Find k so that x  1 is a factor of 2 x3  x 2  3x  k
29.
Given P ( x)  ax 3  bx 2  cx  d with P(3)  0 , P (2)  0 , P (2)  0 and P (0)  24 ,
find the values of a, b, c and d.
36
30.
If f ( x)  x 3  4 x 2  3 x  k and f (1)  4 , find the value of k.
31.
What is the remainder when x5  6 x 4  6 x3  9 x 2  7 x  2 is divided by x 2  4 x  1 ?
32.
What is the remainder when 7 x 20  5 x10  4 x5  3 is divided by x  1 ?
33.
Write a function with integral coefficients and roots of 3  2 , 2i and 3. You may
leave your answer in factored form with only integral coefficients in the factors.
34.
If f ( x)  ax 3  bx 2  cx  d has zeros at 3 and 5, where a, b, c and d are relatively
prime integers (no common factors), find the value of c.
37
35.
Two of the roots of 4 x 3  8 x 2  cx  d  0 are 3 and 1. Find the values of c and d.
36.
Write an equation with relatively prime integral coefficients whose roots are twice those
of 2 x 2  3x  7  0
37.
Write a new equation with relatively prime integral coefficients whose roots are equal to
those of the following equation multiplied by 3 : x3  4 x 2  x  4  0
38.
The equation 4 x 3  3x 2  2 x  k  0 has roots that are opposite those of the equation
2 x3  cx 2  x  5  0 . Find the values of c and k.
39.
The equation 3x3  15 x 2  cx  d  0 has roots that are reciprocals of those of the
equation 2 x 3  4 x 2  5 x  1  0 . Find the values of c and d.
40.
Find the value of k so that the equation 4 x3  kx  1  0 has a double root.
38