Math Notes

1. Read Module 3 Topic 7, Module 4 Topic 1 and 2, Module 5 Topic 1~2.

2. Do the drills for the topics.

3. Read the Chapter 3 sections 2, 5 and Chapter 4 sections 1~3 in your textbook.

4. Do Homework for week 5 (you can find the list in the conference).

Week 5 Supplementary Notes

Chapter 3

Section 3.2: Polynomial Function of Higher Degree

A polynomial function P is given by , where the coefficients are real numbers and the exponents are whole numbers. This polynomial is of nth degree.

Far-Left and Far-Right Behavior

The behavior of the graph of a polynomial function as x becomes very large or very small is referred to as the end behavior of the graph. The leading term of a polynomial function determines its end behavior.

x becomes very large

x → ∞

x becomes very large

x → ∞

x becomes very small

-∞ ← x

x becomes very small

-∞ ← x

We can summarize the end behavior as follows:

The Leading-Term Test

If is the leading term of a polynomial, then the behavior of the graph as x → ∞ or as x → −∞ can be described in one of the four following ways.

If n is even and an >0:

▼ ▼ | If n is even and an <0:▲ ▲ |

If n is odd and an >0: ▲▼ |

If n is odd and an <0: ▲

▼ |

Polynomial Function, Real Zeros, Graphs, and Factors (x − c)

If c is a real zero of a function (that is, f(c)=0), then (c,0) is an x-intercept of the graph of the function.

Some higher degree polynomials can be factored by grouping

Example: Find the zeros of

Solution: We factor by grouping as follows:

Then, by the principle of zero products, the solutions of the equation are 2, -3, and 3. These are the zeros of.

Example: Find the zeros of

Solution: We factor as follows:

We now solve the equation to determine the zeros. We use...