Download CA LEC 3.4 Solutions

Zeros of Polynomials
Section 3.4
Apply the Rational Zero Theorem
Rational Zero Theorem:
If f (x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x + a0 has integer coefficients and an ≠ 0,
and
if
p
(written in lowest terms) is a rational zero of f,
q
then
p is a factor of the constant term a0 and q is a factor of the leading coefficient an.
If a rational zero exists for a polynomial, then it must be of the form:
p
q

Factors of a0 (constant term)
Factors of an (leading coefficient)
1.
List all possible rational zeros of f  x   4 x3 13x2  32x 15 .
2.
Find the zeros and their multiplicities. f  x   2 x4  x3  5x2  x  3 .
3.
Find the zeros and their multiplicities f  x   x3  2 x2  14 x  5
Apply the Fundamental Theorem of Algebra
Fundamental Theorem of Algebra:
If f (x) is a polynomial of degree n ≥ 1 with complex coefficients, then f (x)
has at least one complex zero.
Linear Factorization Theorem:
If f (x) = anxn + an-1xn-1 + an-2xn-2 + … + a1x + a0 where n ≥ 1 and an ≠ 0,
then f (x) = an(x – c1)(x – c2)…(x – cn) where c1, c2, … cn are complex
numbers.
Conjugate Zeros Theorem:
If f (x) is a polynomial with real coefficients and if a + bi (b ≠ 0) is a zero of
f (x) then its conjugate a – bi is also a zero of f(x).
Number of Zeros of a Polynomial:
If f (x) is a polynomial of degree n ≥ 1 with complex coefficients, then f (x)
has exactly n complex zeros provided that each zero is counted by its
multiplicity.
4.
Given that 1  5i is a zero of f  x   x3  3x2  28x  26 find the remaining zeros and factor f  x  as a
product of linear factors.
1
is one solution to 2 x 4  x3  17 x 2  9 x  9  0 , find the remaining solutions.
5.
Given that x  
6.
Find a polynomial f (x) of lowest degree with zeros of 3i and 2 (multiplicity 2).
7.
Find a polynomial p  x  of degree 3 with zeros – 2i and 4. The polynomial must also satisfy the condition
that p  0   32 .
2
Apply Descartes' Rule of Signs
Descartes' Rule of Signs:
Let f (x) be a polynomial with real coefficients and a nonzero constant term. Then,
The number of positive real zeros is either
the same as the number of sign changes in f (x) or
less than the number of sign changes in f (x) by a positive even integer.
The number of negative real zeros is either
the same as the number of sign changes in f (–x) or
less than the number of sign changes in f (–x) by a positive even integer.
8.
Determine the number of possible positive and negative real zeros.
f  x   x5  x4 14x3  32x2  176x  320
Number of possible positive real zeros
Number of possible negative real zeros
Number of imaginary zeros
Total (including multiplicities)
Find Upper and Lower Bounds
A real number b is called an upper bound of the real zeros of a polynomial if all real zeros are less than or equal to b.
A real number a is called a lower bound of the real zeros of a polynomial if all real zeros are greater than or equal to a.
Upper and Lower Bounds
9.
Let f (x) be a polynomial of degree n ≥ 1 with real coefficients and a positive leading
coefficient. Further suppose that f (x) is divided by (x – c).
1.
If c > 0 and if both the remainder and the coefficients of the quotient are
nonnegative, then c is an upper bound for the real zeros of f (x).
2.
If c < 0 and the coefficients of the quotient and the remainder alternate in sign
(with 0 being considered either positive or negative as needed), then c is a lower
bound for the real zeros of f.
Given p  x   6x4 17 x3 12x2  23x  12 , show that 4 is an upper bound and –2 is a lower bound for the
real zeros.