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Algebra II Pre-AP
Rev 2012-13
5.7: Fundamental Theorem of Algebra
“I WILL …
Find all zeros of a polynomial equation.
Write a polynomial equation given some zeros.”
I. Theorems
A. _________________________is the number of solutions in a polynomial
equation with rational coefficients is equal to the degree of the polynomial when
including all complex solutions and solutions of multiplicity. That includes all
____________, ____________, and ____________ roots.
B. _________________________: All complex zeros (imaginaries and radical)
come in conjugate pairs. If a zero is given, use the conjugate and ______
___________________.
C. Every _________________________ or ____________________ Number has
TWO roots
D. To write a polynomial equation:
A. Look and see if there are any imaginary or radical solutions. If there are,
include its conjugate.
B. List all of the factors
C. Sum and product: The complex factors are more difficult to FOIL so
you can use the idea that the complex factors will always make where the
zeros multiply to make c and add to make b
D. FOIL the remaining factors and set to f(x).
II. Finding Zeros
A. Use the rational root theorem (P(x) over Q(x)) to make a list of potential answers.
B. ___________________ the function
C. Do ___________________or ___________________using these zeros to until
you got it down to a ___________________ equation. You can now use the
quadratic formula as well.
IV. Model Problems
Ex 1: Write the simplest
polynomial function with
the given zeros of 4, –1, –2
and highest degree is 3.
Ex 2: Write the simplest
polynomial function with
the given zeros of 2, –2, and
0 and highest degree is 3.
Your Turn: Write the
simplest polynomial
function with the given
zeros of 3, 0, –2 and highest
degree is 3.
Algebra II Pre-AP
Rev 2012-13
Ex 3: Determine the LINEAR FACTORS
that has 3, 1/2, and 3/2 are zeros. No
fractions or decimals are accepted.
Ex 4: Find a second degree
polynomial function with
real coefficients that has √2
is a zero.
Your Turn: Determine the LINEAR
FACTORS that has –2, 1/4, and –1/2 are
zeros. No fractions or decimals are
accepted.
Ex 5: Find a fourth degree
polynomial function with
real coefficients that has –1,
–1, and 3i are zeros.
Your Turn: Find a fourth
degree polynomial function
with the given zeros of +1,
–2 and –2i
Ex 6: Write the simplest function with
zeros of 2 + i and √3
Ex 7: Write the simplest function with
zeros of 2 and 3 + i
Ex 8: Write the simplest function with
zeros of 1 – 3i, 3, and –2
Your Turn: Write the simplest function
with zeros of 1 with a multiplicity of –3
and 2 – i
Algebra II Pre-AP
Rev 2012-13
Ex 9: Find all the zeros of f(x) = x4 – 3x3 + Ex 10: Find all the zeros of f(x) = x4 – 5x3 +
6x2 + 2x – 60 given that 1 + 3i is a zero of f. 4x2 + 2x – 8 given that 1 + i is a zero of f.
Ex 11: Find all the zeros of f(x) = x3 – 4x2 + Your Turn: Find all the zeros of f(x) = x4 –
21x – 34 given that 1 + 4i is a zero of f.
5x3 + 2x2 + 22x – 20 given that 3 – i is a
zero of f.
Assignment: Worksheet