Name: ________________________________________
Date: _____________
DO NOW:
In ABC, the measure of A is 30and the measure of B is 60. If AB is 12 units long, what is the
area, in square units, of ABC?
A. 12
B. 18 3
C. 36
D. 72
E. 72 3
3.4 Zeroes of Polynomials
Objectives: I will be able to:
 Use the Rational Zero Theorem to find possible rational solutions
 Find zeroes of polynomial functions
 Solve polynomial equations
 Use the Linear Factorization Theorem to find polynomials with given zeroes
 Use Descartes’ Rule for signs
To help us understand the next theorem, let’s consider the polynomial P(x) = (x − 2)(x − 3)(x + 4) = x3 − x2 −
14x + 24
From the factored form we see that the zeros of P are 2, 3, and −4. When the polynomial is expanded, the
constant 24 is obtained by multiplying (−2) × (−3) × 4. This means that the zeros of the polynomial are all
factors of the constant term. The following generalizes this observation.
Rational
Zeroes
Theorem:
 If the polynomial P(x) = anxn + an−1xn−1 +···+ a2x2 + a1x + a0
has integer coefficients, then every rational zero of P is of the form
p
where p is a factor of the constant term a0, and q is a factor of the
q
leading coefficient an.
 The Rational Zeros Theorem does NOT list irrational zeros. These will
need to be found using other means like the quadratic formula.
Example 1:
Find the rational zeros of
a) List all possible rational zeros.
b) Use synthetic division to test the possible rational zeros and find an actual zero.
c) Use the quotient from part b to find the remaining zeros of the polynomial function.
Now You Try:
1. Find all zeroes of f (x) = x 3 + x 2 - 5x - 2
2. Find all zeroes of f (x) = x 4 - 5x 3 - 5x 2 + 23x +10
Example 2:
Factor the polynomial P (x) = 2x3 + x2 − 13x + 6, and find all its zeros.
Example 3:
Solve x 4 - 6x 3 + 22x 2 - 30x +13 = 0
Now You Try:
Solve x 3 - 2x 2 -11x +12 = 0
Properties of
Roots of
 Nonreal Zeros: If a complex number a + bi , b ̸= 0, is a zero of a
polynomial function P with real coefficients, then its conjugate, a - bi , is
Polynomial
Equations
also a zero. Hence, for a polynomial with real coefficients, nonreal zeros
occur in conjugate pairs.
 Irrational Zeros: If a + c b , where a,b, and c are rational and b is not a
perfect square is a zero of a polynomial function P with rational
coefficients, the its conjugate, a - c b , is also a zero. Hence, for a
polynomial with rational coefficients, irrational zeros occur in conjugate
pairs.
Fundamental
Theorem of
Algebra:
 Every polynomial function of degree n, with n ≥ 1, has at least one
complex zero. (Remember a real number is a complex number with
imaginary part zero.)
 Every polynomial function f of degree n, with n ≥ 1, can be factored into n
linear factors (not necessarily unique).
Example 4:
Find a third–degree polynomial function f(x) with real coefficients that has – 3 and i as
zeroes such that f(1) = 8.
Now You Try:
1. Find a fourth–degree polynomial function f(x) with real coefficients that has i and 3i as
zeroes such that f(– 1) = 20.
2. Find a polynomial of lowest degree with rational coefficients that has −2i and 1+
some of its zeros.
Descartes’s
Rule of Signs
3 as
Here's a striking theorem due to Descartes in 1637, often known as "Descartes' rule of
signs":
The number of positive real roots of a polynomial is bounded by the number
of changes of sign in its coefficients.
Example 5:
Determine the possible numbers of positive and negative real zeroes of
f (x) = x 4 -14x 3 + 71x 2 -154x +120
a) Count the number of changes in f(x).
Since f(x) has ___________ sign changes, it has ___, ____, or ___ positive real zeroes.
b) Count the number of changes in f(– x).
Now You Try:
Determine the possible numbers of positive and negative real zeroes of
f (x) = x 3 + 2x 2 + 5x + 4 . Then find the zeroes.
SUMMARY:
Assignment:
pg. 386 – 390: # 1 – 11 Odd, 11 – 21 Odd, 27, 29, 33, 37, 39, 43, 47, 51, 91, 98 - 100