Name:
Date:
Advanced Math
Chapter 2 Outline
This test will be given on ________________. The test will cover Chapter 2.1, 2.3, 2.4, 2.5, and
P.6 in our textbook. Part of the test will be non-calculator and part of it will allow for the use of a
calculator.
Topics from these sections include…
Ch. 2.1~Linear and Quadratic Functions with Modeling
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Polynomial Functions
Linear Functions and their graphs
Average rate of change
Linear correlation and Modeling
Quadratic functions and their graphs
Applications of quadratic functions
Ch. 2.3~Polynomial Functions of Higher Degree with Modeling
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Graphs of polynomial functions
End behavior of polynomial functions
Zeros of polynomial functions
Ch. 2.4~Real Zeros of Polynomial Functions
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Long Division of polynomial function
Synthetic Division
The Factor Theorem
The Remainder Theorem
Rational Zeros…
o Potential zeros
o Finding zeros with a calculator
Completely factoring a polynomial
Ch. P.6~Complex Numbers
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Complex Numbers
-1 = i
Powers of i
Z = a + bi and Z = a - bi
Operations with complex numbers
o Addition
o Subtraction
o Multiplication
o Division
 Complex conjugates Z = a + bi and Z = a - bi
 Complex roots of a quadratic
1
Name:
Date:
Advanced Math
Ch. 2.5~Complex Zeros and The Fundamental Theorem of Algebra
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The Fundamental Theorem of Algebra
Linear Factorization Theorem
Complex conjugate zeros
Finding a polynomial from given zeros
Factoring a polynomial with complex zeros
Finding complex zeros
Factoring with real number coefficients
o Factoring a polynomial…also know that odd degree polynomials have at
least one real zero.
2
Name:
Date:
Advanced Math
Chapter 2 - Practice Problems
Please note: These problems are collected from previous tests and quizzes on this
material
1.
Use Long Division to divide f(x) by d(x) and write an answer in polynomial form.
f(x) = 5x4 + 14x3 + 9x ;d(x) = x2 + 3x
2.
Use long division to find the quotient and remainder when dividing
-2x 3 + 5x 2 - 6x + 5 by 2x-1
3.
Use the factor theorem to determine which of the following are factors of
f ( x) = 3x 4 -17x 3 - 27x 2 +173x + 60
x+2, x-3, x+4, x-5
4.
Use the remainder theorem to find the remainder when f(x) is divided by x – k.
Check your answer by using synthetic division.
f(x) = 3x3 - 2x2 + x – 5 ;k = -2
5.
Use the rational zero theorem to come up with a list of potential rational zeros.
Then determine which ones, if any are zeros.
f(x) = 2x4 - x3 - 4x2 - x – 6
6.
Find all of the real zeros of the function, finding exact values whenever possible.
Identify each zero as rational or irrational.
f(x) = x4- x3- 7x2 + 5x + 10
7.
Use long division to find the quotient and remainder when
(4x 3 + 8x 2 - 9x -18) ¸ ( x + 2)
8.
Use synthetic division to find the quotient and remainder when
(3x
9.
3
-16x 2 - 72) ¸ ( x + 4)
(
)
Find the remainder of x 3 - 3x 2 + 5 ¸ ( x + 3) without doing long or synthetic
division.
3
Name:
Date:
10.
Advanced Math
Use the factor theorem to determine which of the following are factors of
x 3 - 5x 2 + 3x + 9. Work must be shown to justify your answer
a) x -1
b) x + 3
c) x - 3
11.
Find the remaining zeros of x3 – 8x2 + 9x + 6 if x = 2 is itself a zero.
In problems 12-16 perform the indicated operation (without using a calculator) and write
the result in a + bi form.
12.
(-5 + 7i) + (14 + 3i)
13.
(-3 + 2i)( 4 - i)
14.
1+ 3i
5 - 2i
15.
i 33 + i15
16.
i 23 - i 18
i 57
17.
If z = -3-5i, then find
18.
If z = 5-12i, the find z · z
19.
Solve for x and y to make the equation true.
z·z
(3 – 2i) – 8 = x – (-7 + yi)
20.
Simplify (i 29 )
71
4
Name:
Date:
Advanced Math
-12
and then
-3
(
-12
)( -3)
21.
Simplify
22.
I should have used pencil! Your pen just exploded
all over the graph in figure below. You remember
y
that the function is 2x 3 - 5x 2 - 4x + 3. Use what’s
visible below and synthetic division to help you
come up with a completely factored
4
polynomial.
–3 –2 –1
1
2
3 x
–4
f (x ) = 2 x 3 - 5x 2 - 4 x + 3
5
Name:
Date:
Advanced Math
23. The graph and some additional information about a polynomial P(x) are presented
below.
• P(x) is a degree 5 polynomial with real
coefficients.
• The graph of P(x) for real numbers x is
given on the grid. The only x-intercepts of
P are at x = 4 and x = –3.
• In the complex numbers, P(i+1) = 0.
Find the polynomial P(x)
Your final answer should be expressed as linear
and irreducible quadratic factors with real
coefficients.
24. a. Find all solutions of the equation
b. Finding the solutions of the equation
cube roots of what number?
.
is equivalent to finding the three
6
Name:
Date:
Advanced Math
25. You are given this information about a polynomial Q(x):
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Q(x) is a degree 5 polynomial with real coefficients.
The graph of Q(x) for real numbers x is given on the
grid. Its only x-intercepts are at x = –2 and x = 0.
Q(–1) = 4.
In the complex number system, Q(1 – 2i) = 0.
Find the factorization of Q(x) in the real number
system.
26.
Write a polynomial function in standard form (meaning it has real coefficients)
whose degree is 3 and has zeros at 2 and 1 + i.
7
Name:
Date:
Advanced Math
Practice Problems Solutions
1.
Quotient: 5x 2 - x + 3, remainder: 0
2.
Quotient: -x 2 + 2x - 2, remainder: 3
3.
x-5
4.
-39
1
3
5.
±1,± ,±2,±3,± ,±6
2
2
6.
x = -1 rational
x = 2 rational
x =  5 irrational
7.
Quotient: 4x 2 - 9, remainder: 0
8.
Quotient: 3x 2 - 28x +112, remainder: -520
9.
-49
10.
x -3
x  2, x  3  2 3
11.
12.
9 + 10i
13.
-10 + 11i
-1+ 17i
14.
29
15.
0
16.
-1-i
17.
34
18.
169
19.
x = -12 and y = 2
20.
-i
21.
2, and -6
22.
( x +1)(2x -1)( x - 3)
1
2
23.
P( x ) = - ( x + 3) ( x - 4 ) x 2 - 2x + 2
36
3  3i 3
x
, x  3
24a.
2
24b. -27
1
2
25.
Q( x ) = - x ( x + 2) x 2 - 2x + 5
2
3
2
26.
x - 4x + 6x - 4
(
(
)
)
8