1. Chapter 6
Review

2. Main Concepts Properties of Exponents
Determining End Behavior of Polynomial Functions
Adding, Subtracting, and Multiplying Polynomials
Finding Zeros of Polynomial Functions
Descartes’ Rule of Signs
Making P/Q values
Long and synthetic division
Graphing Polynomial Functions
End behavior, multiplicity rule
Factor by Grouping and u substitution

3. Properties of Exponents2 2

4. End Behavior
End behavior is how each polynomial behaves as it approaches ∞ or -∞ .
The is the value that determines whether the graph is positive or negative, while the exponent (5) is the value that determines whether the arms of the graph will go the same or different ways. In this case, the graph is negative because -2 is negative, and the arms go opposite ways because the exponent is odd. If the exponent had been even, both arms would face the same way.

5. Coefficient is pos. Coefficient is neg. Coefficient is pos
Coefficient is pos Coefficient is neg Coefficient is pos Coefficient is neg.
Exponent is even Exponent is even Exponent is odd Exponent is odd

6. Adding, Subtracting, Multiplying Polynomials
Add: (x + 3) + (2x - 4) Answer: 3x - 1
Subtract: (3x - 7) - (x - 1) Answer: 2x - 6
Multiply: (3x - 1)(x - 4)(x + 1) Answer: (see below)

7. Factoring by grouping, u substitution
To factor by grouping, find a common denominator comprised of two terms in a four-term equation. For example,
can be factored by grouping first into x^2(x-2) + 4(x-2), then finally into (x^2+4)(x-2).
To u substitute, replace with u and then solve like a quadratic. For example,
can be substituted into u^2 +3u + 2, then into (u+2)(u+1), then back into (x^2+2)(x^2+1).

8. Synthetic Division tested zero (n) leading coefficient (a)
coefficient of second term (b)
coefficient of third term (c)
coefficient of fourth term (d)
a * n
(b + (a * n)) * n
... a
b + (a * n)
((b + (a*n))*n) + c

9. Ex.
Determine whether or not x = 2 is a zero of the following polynomial:

10. Polynomial Long Division

11. Descartes’ Rule of Signs
Descartes’ Rule of Signs states that:
The number of positive solutions of a polynomial is equal to the number of sign changes in f(x) or the number of sign changes minus an even number. (ex. If there are 7 sign changes, there are either 7, 5, 3, or 1 solution(s))
The number of negative solutions of a polynomial is equal to the number of sign changes in f(-x) or the number of sign changes minus an even number.
The other solutions are imaginary.

12. + - i 3 1 2
Sign changes: 3
Sign changes: 1

13. P/Q Values P Q
Divisor of P / Divisor of Q = Possible polynomial solution.

14. Multiplicity Rule
Multiplicity is when a zero occurs more than once in a polynomial.
If a zero is repeated an odd number of times, the graph will cross the X-Axis at that zero. If a zero is repeated an even number of times, the graph will not cross the X-axis at that zero and rather become a turning point.

15. Put these concepts together!
Graph:

17. Jeopardy: https://www.playfactile.com/chaptersixmath/play