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Algebra 2: Unit 4
Name_______________________________
Hour__________
Unit 4 Notes:
Complex numbers and
Higher Degree Polynomials
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Algebra 2: Unit 4
4.1 Solve Quadratic Equations with Imaginary Solutions
Vocabulary:

The imaginary unit i is defined as ___________________.
o Note then, that ______________.

A _____________number written in standard form is
_____________. Where a is the ___________ part and bi is the
_______________ part.
Solving Quadratic Equations. State if the solutions are real or imaginary:
Solve the equation.
1. x2 = - 81
2. 2r2 = 24
3. z2 + 6 = 4
4. 3p2 + 7 = -9p2 + 4
5. x2 + 15 = - 35- x2
6. 3x2 + 13 = - 23
Adding and Subtracting:
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Algebra 2: Unit 4
7. x2 – 3x + 5 = 0
8. 2x2 – 1 = 3x + 4
9. x2 – 6x = -9
10. 3x2 – 3x + 4=0
Summary
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Algebra 2: Unit 4
4.2 Perform Operations with Complex Numbers
Remember i=_________
i  i  i 2 , i 2  1
Simplify these imaginary numbers:
i2
i3
i4
i5
i31
i42
ADDING AND SUBTRACTING COMPLEX NUMBERS
Write the expression as a complex number in standard form.
Standard Form _____________________
1. (6 – 3i) + (5 + 4i)
2. (8 + 20i) – (8 – 9i + 7)
3. (3 – 2i) + (6 – 3i)
4. (i3 + 1 + 8i) – (7 + 5i)
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Algebra 2: Unit 4
MULTIPLYING AND DIVIDING COMPLEX NUMBERS
Write the expression as a complex number in standard form.
5. 6i(3 + 2i)
6. (-1 – 5i)(- 1 + 5i)
7. 3i(2 - i)(4 + 2i)
8. (2 + i)(6 - 5i) + (-4 + 2i)
Complex Conjugates: (Note, a + bi and a – bi are complex conjugates)
Write the expression as a complex number in standard form. To do this, multiply
the numerator and the denominator by the ___________________.
9.
6i
3i
11.
6  4i
2i
5
10.
4  9i
2  3i
12.
7i
i
Algebra 2: Unit 4
13. Challenge: Simplify the following:
Summary
4.3 Factoring 3rd and 4th degree Polynomials
Factoring we have already done:
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Algebra 2: Unit 4
Factoring 3rd and 4th degree with a GCF
Factor the following 3rd and 4th degree polynomials.
1. 𝑥 3 − 16𝑥 2 − 17𝑥
2. 6𝑥 4 − 4𝑥 3 − 16𝑥 2
3. 12𝑥 4 − 27 = 0
4. 7𝑥 4 −6𝑥 3 −𝑥 2
Factoring by Grouping:
5. y3 – 7y2 + 4y – 28
6. 2𝑝3 + 5𝑝2 + 6𝑝 + 15
Factor a quartic trinomial:
7. u4 + 7u2 + 6
8. x4 – 20x2 + 36
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Algebra 2: Unit 4
Solve by factoring:
9. 3x4 + 15x2 = 18x3
10. 16g4 – 625 = 0
11. 𝑥 4 − 4 = 0
12. 2𝑥 3 − 9𝑥 2 = 10𝑥
Summary
4.4 Long and Synthetic Division of Polynomials and the
Factor Theorem
Vocabulary:
5 divided by 4 is 1 ¼
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Algebra 2: Unit 4
Long Division with polynomials
1. Divide 3x4 – 5x3 + 4x – 6
by x2 – 3x + 5
2. Divide x3 + 3x2 – 7 by x2 – x – 2
3. Divide ( x 2  x  17)  ( x  4)
4. Divide 𝑥 3 + 3𝑥 2 − 7 by 𝑥 2 − 𝑥 − 2
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Algebra 2: Unit 4
5. Divide 2𝑥 4 + 𝑥 3 + 𝑥 − 1 by 𝑥 2 + 2𝑥 − 1
6. Divide (8𝑥 2 + 34𝑥 − 1) ÷ (4𝑥 − 1)
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Algebra 2: Unit 4
7. Divide 2𝑥 2 − 7𝑥 + 10 by 𝑥 − 5
Using Long Division:
Using Synthetic Division:
Can be used to divide any polynomial by a divisor of the form x – k
8. Divide 𝑥 3 − 5𝑥 2 − 2 by 𝑥 − 4
9. Divide 𝑥 4 + 4𝑥 3 + 16𝑥 − 35 by 𝑥 + 5
10. Divide (2 x 3  x 2  8x  5)  ( x  3) using synthetic division
******Factor theorem: A polynomial has a factor (x – k) if and only if the
remainder is zero.***********
Determine if the binomial is a factor of the polynomial.
11.
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Algebra 2: Unit 4
12.
Summary
4.5 Completely Factor Polynomials and Find Zeros
Factor:
Given the polynomial and a factor, factor completely.
1. f ( x)  x 3  10 x 2  19 x  30; ( x  6)
2. f ( x)  3x 3  4 x 2  28x  16; ( x  2)
3. f ( x)  2 x 3  15x 2  34 x  21; ( x  1)
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Algebra 2: Unit 4
Find Zeros:
4. One zero of f ( x)  x 3  2 x 2  23x  60 is 3. What are the other zeros?
5. One zero of f ( x)  10 x 3  81x 2  71x  42 is 7. What are the other zeros?
6. One zero of f ( x)  x 3  12 x 2  35x  24 is 3. Find the other zeros.
Summary
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Algebra 2: Unit 4
4.6 Graphing Polynomial Functions
CUBIC FUNCTIONS (3rd degree)
The graph of a cubic polynomial has the basic property that the graph can cross
the x-axis in at most 3 points. It may cross in fewer points, or it may not cross at
all.
Graph of f(x) = ax3 + bx2 + cx + d
a is positive
a is negative

If the degree of a polynomial function is odd, then the left end of
the graph and the right end of the graph are in
_____________directions.
QUARTIC FUNCTIONS (4th degree)
The graph of a quartic polynomial can cross the x-axis in at most 4 points. It may
cross in fewer points, or it may not cross at all.
Graph of f(x) = ax4 + bx3 +cx2 + dx + e
a is positive
a is negative

If the degree of a polynomial function is even, then the left end
of the graph and the right end of the graph in the __________
direction.
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Algebra 2: Unit 4
State the degree of the function and if a is positive or negative
1.
2.
3.
4.
Zeros and Crossing x-axis
A function of nth degree can cross the x-axis at most n times. That
means it can have at most n real zeros. If it doesn’t cross n times,
some solutions are just imaginary then.
5. How many times can 𝑓(𝑥) = 𝑥 7 − 𝑥 3 + 7𝑥 − 1 cross the x-axis?
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Algebra 2: Unit 4
THE MULTIPLICTY OF ZEROS
State the zeros of the graph and the multiplicity of them. Also state if the
polynomial function is of even or odd degree at least what degree it is.
6.
7.
8.
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Algebra 2: Unit 4
Sketching Graphs
Sketch a rough graph of the polynomial functions based on the
degree, value of a, and the zeros and their multiplicity.
9. 𝒚 = (𝒙 + 𝟓)(𝒙 − 𝟏)𝟐 (𝒙 − 𝟕)
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
10. 𝒇(𝒙) = (𝒙 − 𝟑)(𝒙 − 𝟏)(𝒙 + 𝟏)(𝒙 + 𝟐)
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
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Algebra 2: Unit 4
11. 𝒚 = −𝟐(𝒙 − 𝟑)𝟐 (𝒙 + 𝟒)
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
12. 𝒚 = (𝒙 − 𝟏)(𝒙 + 𝟏)(𝒙 − 𝟓)
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
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Algebra 2: Unit 4
13. 𝒇(𝒙) = −(𝒙 + 𝟑)𝟑
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
14. 𝒚 = −𝟒(𝒙 − 𝟏)(𝒙 + 𝟏)𝟐 (𝒙 + 𝟔)
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
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Algebra 2: Unit 4
15. 𝒚 = −𝒙𝟒 − 𝒙𝟑 + 𝟐𝒙𝟐
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
16. 𝒚 = 𝒙𝟑 +𝟏𝟗𝒙𝟐 + 𝟏𝟐𝟎𝒙 + 𝟐𝟓𝟐 given (𝒙 + 𝟔) is a factor.
Degree: even or odd
a: positive or negative
zeros and multiplicity of each:
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Algebra 2: Unit 4
Summary
4.7 Write Equations of Polynomial Functions from a Graph
Steps to writing an equation from a graph:
1) Locate the ________________ or ________________.
2) Determine the ___________________ of each zero.
3) Write the polynomial in _________________ form.
4) Use a point on the graph to determine the value of ____. If possible, use
the __-_______________ for your point.
Write equations from the graphs:
1.
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Algebra 2: Unit 4
2.
3.
4.
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Algebra 2: Unit 4
5.
6. Write an equation for a polynomial function with the following zeros and point
on the graph.
Zeros: 2 (multiplicity 1), -1 (multiplicity 1) and 4 (multiplicty 2).
Point on Graph (y-intercept): (0, 2)
7. Write an equation for a polynomial function with the following zeros and y-int.
Zeros: -3 (multiplicity 2) and 1 (multiplicity 1)
Point on Graph: (2, -7)
Summary
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