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Using Our Tools to Find the Zeros
of Polynomials
17 November 2014
Using Our Tools to Find the Zeros of
Polynomials
Do you feel like this?
Fundamental Theorem
of Algebra
The Rational Root Theorem
Integral Root Theorem
Quadratic Formula
Descartes’ Rule of Signs
Remainder Theorem
Location Principle
Synthetic Division
Factor Theorem
Upper Bound Theorem
Lower Bound Theorem
Using Our Tools to Find the Zeros of
Polynomials
Let organize our thinking.
Fundamental Theorem
of Algebra
The Rational Root Theorem
Integral Root Theorem
Quadratic Formula
Descartes’ Rule of Signs
Remainder Theorem
Location Principle
Synthetic Division
These are all tools to help
us solve polynomials
Factor Theorem
Upper Bound Theorem
Lower Bound Theorem
Using Our Tools to Find the Zeros of
Polynomials
• Review
– Fundamental Theorem of Algebra
• Every polynomial equation with a degree greater than
zero has a least one root in the set of complex numbers
• Based on the corollary, a polynomial equation of
degree n has exactly n complex roots.
– Quadratic Formula
−𝑏± 𝑏2 −4𝑎𝑐
2𝑎
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
– Remainder Theorem
• If a polynomial P(x) is divided by x-r, the remainder is a constant P(r)
and P(x) = (x-r)(Q(x)) + P(r) where Q(x) is one degree less than the
degree P(x)
– Factor Theorem
• This means that we can try to find where P(r) = 0. In this case x-r is a
factor. Also Q(x) may be factorable.
– Synthetic Division
• Shorthand process to divide P(x) by x-r .
• Process is more efficient to try potential zeros (roots)
• synthetic division is an easy way to get a depressed polynomial. (In
the case of a degree 3, finding one root opens up the use of factoring
or the quadratic formula on the resulting degree 2 so the process is
shortened considerably)
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
– Rational Root Theorem - If a polynomial function,
written in descending order of the exponents, has
integer coefficients, then any rational zero must
be of the form ± p/ q, where p is a factor of the
constant term and q is a factor of the leading
coefficient.
• This means that we can very quickly create a “short” list
of possible solutions
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
• Descartes’ Rule of Signs
– A method of determining the maximum number of
positive and negative real roots of a polynomial.
– For positive roots, start with the sign of the coefficient
of the lowest (or highest) power. Count the number of
sign changes n as you proceed from the lowest to the
highest power (ignoring powers which do not appear).
Then n is the maximum number of positive roots.
Furthermore, the number of allowable roots is n, n-2,
n-4, ....
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
• Descartes’ Rule of Signs
– A method of determining the maximum number
of positive and negative real roots of a polynomial.
– For negative roots, starting with a polynomial f(x),
write a new polynomial f(-x) with the signs of all
odd powers reversed, while leaving the signs of
the even powers unchanged. Then proceed as
before to count the number of sign changes n.
Then n is the maximum number of negative roots.
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
– Location Principle
• if a continuous function has opposite signs for two
values of the independent variable, then it is zero for
some value of the variable between these two values.
– Upper Bound Theorem
• If c is a positive real number and P(x) is divided by x-c
and the resulting quotient and remainder have no
change in sign, the P(x) has no real zero greater than c.
Thus c is the upper bound of the zeros of P(x)
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
– Location Principle
• if a continuous function has opposite signs for two
values of the independent variable, then it is zero for
some value of the variable between these two values.
– Lower Bound Theorem
• If c is an upper bound of the zeros of P(-x), then –c is a
lower bound of the zeros of P(x).
Using Our Tools to Find the Zeros of
Polynomials
• Review (Cont’d)
– Multiplicity - How many times a particular
number is a zero for a given polynomial. For
example, in the polynomial function f(x) = (x –
3)4(x – 5)(x – 8)2, the zero 3 has multiplicity 4, 5
has multiplicity 1, and 8 has multiplicity 2.
Although this polynomial has only three zeros, we
say that it has seven zeros counting multiplicity.
Using Our Tools to Find the Zeros of
Polynomials
• Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
• Step 1 - Use Rational Root Theorem
+ 1, + 2, + 3, + 5, + 6, + 10, + 15, + 30
• Step 2 - Use Descartes Rule of Signs
Pos – 0
Neg - f(-x) = x4 - 9x3 + 31x2 - 49x + 30 = 0
Therefore - 4, 2, or 0 negatives
Sometimes it helps to make a chart.
Using Our Tools to Find the Zeros of
Polynomials
• Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
• Step 3 - Use Synthetic Division. Since there are only
negative roots, we will start by testing negative possible
zeros.
f(-2)
-2
1
1
9
-2
7
31
-14
17
49
-34
15
30
-30
0
Using Our Tools to Find the Zeros of
Polynomials
• Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
• Step 3 - Use Synthetic Division. Since there are only
negative roots, we will start by testing negative possible
zeros.
f(-2)
-2
1
1
9
-2
7
31
-14
17
49
-34
15
x3 + 7x2 - 17x + 15 = 0
• Follow same process for x3 + 7x2 - 17x + 15 = 0
• Try f(-3)
30
-30
0
Using Our Tools to Find the Zeros of
Polynomials
• Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
Follow same process for x3 + 7x2 - 17x + 15 = 0
• Try f(-3) x2 - 4x + 5 = 0
(note: we now have 2 of 4 zeros)
• Step 4 - Use Quadratic Formula
𝑥=
−4± −4
2
These will be imaginary.
Using Our Tools to Find the Zeros of
Polynomials
Homework - Find all zeros of the following:
• x3 -4x2 + x + 2 = 0
• x4 + 3x2 – 4 = 0
• x3 + 3x2 – 2x – 8 = 0
• 2x3 + 7x2 + 7x + 2