Download Section 4.4: Theorems About Roots of Polynomials

Section 4.4: Theorems About Roots of Polynomials
March 06, 2015
Section 4.4: Roots of Polynomials
Irrational Root Theorem
Let a and b be rational numbers and let √b be an irrational
number. If a + √b is a root of a polynomialequation with
rational coefficients, then the conjugatea - √b also is a root.
Example:
A polynomial equation with integer coefficients has the
roots 1 +√3 and -√11. Find two additional roots.
Imaginary Root Theorem
If the imaginary number a + bi is a root of a
polynomial equation with real coefficients, then
the conjugate a - bi also is a root.
Example:
A polynomial equation with integer coefficients has the
roots 3 - i and 2i. Find two additional roots.
Section 4.4: Theorems About Roots of Polynomials
March 06, 2015
State the number of complex zeros and the possible number of
real and imaginary zeros for each function.
1.) f(x) = x4 ­ 9x2 + 18
2.) f(x) = 27x6 + 208x 3 ­ 64
# of complex = # of complex = # of real = # of real = # of imaginary = # of imaginary = 3.) f(x) = 5x5 + 36x3 + 7x
4.) f(x) = 27x9 + 8x6 ­ 27x3 ­ 8
# of complex = # of complex = # of real = # of real = # of imaginary = # of imaginary = Rational Root Theorem
For the polynomial...
f(x) = anxn + an-1 xn-1 +...+ a1 x + a0
the possible rational roots are±p/q
p = factors of a0 (constant/number term)
q = factors of an (leading coefficient)
This is how we can find roots without a graphing
calculator!
Section 4.4: Theorems About Roots of Polynomials
March 06, 2015
List all possible rational roots.
1.) 6x3 + 11x2 - 3x - 2 = 0
2.) x3 + 8x2 +16x + 5 =0
3.) 2x4 - x3 + 7x2 -9x -18 =0
Descartes' Rule of Signs
For the polynomial P(x) the number of...
Positive Real Roots = # of sign changes or less that byan
even number(aka subtract 2 until you reach 1 or 0)
Negative Real Roots = #of sign changes for P(-x) orless
that by an even number
*will change the sign in front of any odd power of x
The reason it is "or less that by an even number" is because
if a root is not real, then it must be imaginary and imaginary
roots always come in pairs
Section 4.4: Theorems About Roots of Polynomials
List the possible number of positive roots.
1.) x3 + 6x2 +10x + 3 = 0
possible number of positive roots: __________
2.) x4 +3x2 - 8 = 0
possible number of positive roots: __________
3.) 2x5 - 6x3 + 6x2 + 3 = 0
possible number of positive roots: __________
List the possible number of negative roots.
1.) x3 + 6x2 +10x + 3 = 0
possible number of negative roots: __________
2.) x4 +3x2 - 8 = 0
possible number of negative roots: __________
3.) 2x5 - 6x3 + 6x2 + 3 = 0
possible number of negative roots: __________
March 06, 2015