1. Zeros of Polynomial Functions
Advanced Math
Section 3.4

2. Number of zeros
Any nth degree polynomial can have at most n real zeros
Using complex numbers, every nth degree polynomial has precisely n zeros (real or imaginary)
Advanced Math

3. Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n, where n > 0,
then f has at least one zero in the complex number system
Advanced Math

4. Linear Factorization Theorem
If f(x) is a polynomial of degree n, where n > 0,
then f has precisely n linear factors
Advanced Math

5. Linear Factorization Theorem applied
Advanced Math

6. Example
Find all zeros
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7. Rational Zero Test
Advanced Math

8. Using the rational zero test
List all rational numbers whose numerators are factors of the constant term and whose denominators are factors of the leading coefficient
Use trial-and-error to determine which, if any are actual zeros of the polynomial
Can use table on graphing calculator to speed up calculations
Advanced Math

9. Example Use the Rational Zero Test to find the rational zeros
Advanced Math

10. Using synthetic division
Test all factors to see if the remainder is zero
Can also use graphing calculator to estimate zeros, then only check possibilities near your estimate
Advanced Math

11. Examples
Find all rational zeros
Advanced Math

12. Conjugate pairs If the polynomial has real coefficients,
then zeros occur in conjugate pairs
If a + bi is a zero, then a – bi also is a zero.
Advanced Math

13. Example:
Find a fourth-degree polynomial function with real coefficients that has zeros -2, -2, and 4i
Advanced Math

14. Factors of a Polynomial
Even if you don’t want to use complex numbers
Every polynomial of degree n > 0 with real coefficients can be written as the product of linear and quadratic factors with real coefficients, where the quadratic factors have no real zeros
Advanced Math

15. Quadratic factors
If they can’t be factored farther without using complex numbers, they are irreducible over the reals
Advanced Math

16. Quadratic factors
If they can’t be factored farther without using irrational numbers, they are irreducible over the rationals
These are reducible over the reals
Advanced Math

17. Finding zeros of a polynomial function
If given a complex factor
Its conjugate must be a factor
Multiply the two conjugates – this will give you a real zero
Use long division or synthetic division to find more factors
If not given any factors
Use the rational zero test to find rational zeros
Factor or use the quadratic formula to find the rest
Advanced Math

18. Examples Use the given zero to find all zeros of the function
Advanced Math

19. Examples
Find all the zeros of the function and write the polynomial as a product of linear factors
Advanced Math

20. Descartes’s Rule of Signs
A variation in sign means that two consecutive coefficients have opposite signs
For a polynomial with real coefficients and a constant term,
The number of positive real zeros of f is either equal to the number of variations in sign of f(x) or less than that number by an even integer
The number of negative real zeros is either equal to the variations in sign of f(-x) or less than that number by an even integer.
Advanced Math

21. Examples Determine the possible numbers of positive and negative zeros
Advanced Math

22. Upper Bound Rule If what you try isn’t a factor, but
When using synthetic division
If what you try isn’t a factor, but
The number on the outside of the synthetic division is positive
And each number in the answer is either positive or zero
then the number on the outside is an upper bound for the real zeros
Advanced Math

23. Lower Bound Rule If what you try isn’t a factor, but
When using synthetic division
If what you try isn’t a factor, but
The number on the outside of the synthetic division is negative
The numbers in the answer are alternately positive and negative (zeros can count as either)
then the number on the outside is a lower bound for the real zeros
Advanced Math

24. Examples
Use synthetic division to verify the upper and lower bounds of the real zeros
Advanced Math

25. Mathematical Modeling and Variation
Advanced Math
Section 3.5

26. Two basic types of linear models
y-intercept is nonzero
y-intercept is zero
Advanced Math

27. Direct Variation Linear k is slope y varies directly as x
y is directly proportional to x
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28. Direct Variation as an nth power
y varies directly as the nth power of x
y is directly proportional to the nth power of x
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29. Inverse Variation Hyperbola (when k is nonzero)
y varies inversely as x
y is inversely proportional to x
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30. Inverse Variation as an nth power
y varies inversely as the nth power of x
y is inversely proportional to the nth power of x
Advanced Math

31. Joint Variation Describes two different direct variations
z varies jointly as x and y
z is jointly proportional to x and y
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32. Joint Variation as an nth and mth power
z varies jointly as the nth power of x and the mth power of y
z is jointly proportional to the nth power of x and the mth power of y
Advanced Math

33. Examples
Find a math model representing the following statements and find the constants of proportionality
A varies directly as r2.
When r = 3, A = 9p
y varies inversely as x
When x = 25, y = 3
z varies jointly as x and y
When x = 4 and y = 8, z = 64
Advanced Math