Problem (2.4)
Name: ____________________
Find the zeros of the polynomial. No calculator.
f(x) = x3 - 5x2 + 3x + 1
Problem (2.4)
Name: ____________________
Find the zeros of the polynomial. No calculator.
f(x) = x3 - 5x2 + 3x + 1
p
p = +- 1
q=+
- 1
1
1
1
q
= +- 1
x2 - 4x - 1 = 0
x=
4 +- 16-4(1)(-1)
2(1)
-5
3
1
1
-4
-1
-4
-1
0
x = 4 +- 20
2
x = 4 +- 2 5
2
x = 2 +- 5
x=1
x = 2 +- 5
(2.5) Complex Zeros
Objective: To find all zeros (including complex zeros) of a
polynomial function and be able to factor it with real coefficients.
Why: Completes the study of zeros of polynomials with real
coefficients.
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
Fundamental Theorem of Algebra:
A polynomial of degree n has n complex zeros (real
and nonreal), some may be repeated.
Linear Factorization Theorem:
A polynomial of degree n has n linear factors,
f(x) = a(x-z )(x-z )...(x-z
), some
factors may
1
2
n be
repeated.
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
Write a polynomial in standard form and identify the
zeros and x-intercepts of the function.
f(x) = (x-3)(x-2i)(x+2i)
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
If a polynomial has nonreal zeros, then the complex zeros
will occur in pairs and be conjugates of one another.
Quad. Form:
x = -b ± b2-4ac
2a
If b2 - 4ac is negative, then there will be 2
complex conjugate zeros.
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
Write a polynomial of minimum degree in
standard form with real coefficients whose zeros
include -3,4, and 2-i.
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
5
4
3
2
Find the zeros of f(x)=x - 3x - 5x + 5x - 6x + 8
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
Find the remaining zeros of
f(x) = 4x4 + 17x2 + 14x +65
if 1-2i is one zero.
Find the zeros.
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
f(x) = 3x5 -2x4 + 6x3 - 4x2 - 24x + 16
Write f(x) as a product of linear and irreducible quadratic
factors, each with real coefficients.
Obj: To find all zeros (including complex zeros) of a polynomial
function and be able to factor it with real coefficients.
HW:
HR: (2.5) Pg. 215: 1, 4, 5, 7, 9, 13, 17-21, 23, 37, 39, 35, 39, 53-55