1. Using Our Tools to Find the Zeros of Polynomials
17 November 2014

2. 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
Upper Bound Theorem
Factor Theorem
Lower Bound Theorem

3. 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
Upper Bound Theorem
Factor Theorem
Lower Bound Theorem

4. 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± 𝑏 2 −4𝑎𝑐 2𝑎

5. 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)

6. 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

7. 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, ....

8. 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.

9. 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) If

10. 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). If

11. 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. If

12. Using Our Tools to Find the Zeros of Polynomials
Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
Step Use Rational Root Theorem
+ 1, + 2, + 3, + 5, + 6, + 10, + 15, + 30
Step 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.

13. Using Our Tools to Find the Zeros of Polynomials
Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
Step Use Synthetic Division. Since there are only negative roots, we will start by testing negative possible zeros.

14. Using Our Tools to Find the Zeros of Polynomials
Find all the zeros of x4 + 9x3 + 31x2 + 49x + 30 = 0
Step Use Synthetic Division. Since there are only negative roots, we will start by testing negative possible zeros.
x3 + 7x2 - 17x + 15 = 0
Follow same process for x3 + 7x2 - 17x + 15 = 0
Try f(-3)

15. 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 Use Quadratic Formula
𝑥= −4± − These will be imaginary.

16. 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