# 6 5 Theorems About Roots of Polynomials Imaginary Roots

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Polynomial Equations
Imaginary Roots
POLYNOMIALS and THEOREMS
Theorems of Polynomial Equations
•    There are 4 BIG Theorems to know
1)   Rational Root Theorem
2)   Irrational Root Theorem
3)   Imaginary Root Theorem
4)   Descartes Rule
1. For polynomial has roots 5 and 3i
- 3i                          3
Other roots ______ Degree of Polynomial ______
Why not -5 ? For real numbers we do not       Why a third degree
need to worry about the conjugate. Only the   polynomial?
complex part has a second root.
(x-3i)(x+3i)(x-5)
2. For polynomial has roots -5i, 4 – 2i
5i , 4 + 2i                      4
Other roots __________ Degree of Polynomial ______
Write a polynomial given the roots 4, 2i, -√5

•   Other roots are -2i and √5
•   Put in factored form
•   y = (x – 4)(x + 2i)(x – 2i)(x + √5)(x – √5)
•   Decide what to FOIL first
y = (x – 4)(x + 2i)(x – 2i) (x + √5)(x – √5)

x      2i             X      +√5
x    X2      2X i     x    X2      X √5
-4i2
-2X i                 -X √5   -5
-2i              =4   -√5

(x² +4)                (x² – 5)
y = (x – 4)(x4 – 5x² + 4x² – 20)
y = (x – 4)(x4 – x² – 20)
BOX it!!!

y = (x – 4)(x4 – x² – 20)

x2        4                               x4       -x2     -20
x2      X4        4x2                       x    X5       -X3       -20X

-5X2      -20                            -4X 4     4x2      80
-5                                         -4

(x4 – 5x² + 4x² – 20)              y = x5 – 4x4 – x3 + 4x² – 20x + 80
y = (x4 – x² – 20)
Write a polynomial given the roots
1+i       Note: When you box or
foil complex numbers,
•   Other root is 1 – i                 you will not have any
imaginary numbers left
•   Put in factored form                in the answer. If you do,
go back and check your
•   y = (x – (1 + i))(x – (1 – i))      arithmetic.

•   Distribute the negatives
X          –1          –i
•   y = (x – 1 – i )(x – 1 + i)
X X²           -x         -ix
•   FOIL or BOX to finish it up
-1 - x         1             i
•   y = x² – 2x + 2
i   ix        -i         -i²=1
Let’s Try One
• Find a fourth-degree polynomial equation
with integer coefficients that has the given
numbers as roots:

3 + i and -2i

(Note: this is problem #29 on tonight’s
homework)

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