Polynomial and Linear Factors Section 6.2 Polynomial and Linear Factors

Example 1 Write each expression as a polynomial in standard form. (𝒙+𝟏)(𝒙+𝟐)(𝒙+𝟑)

Example 2 Write each expression as a polynomial in standard form. (𝒙+𝟏)(𝒙+𝟏)(𝒙+𝟐)

Example 3 Write each polynomial in factored form. 𝟐 𝒙 𝟑 +𝟏𝟎 𝒙 𝟐 +𝟏𝟐𝒙

Example 4 Write each polynomial in factored form. 𝟔 𝒙 𝟑 −𝟏𝟓 𝒙 𝟐 −𝟑𝟔𝒙

Factored Form of a Polynomial Definition 1: The maximum values is the greatest value of the points in a region of the graph. It is called a relative maximum. Similarly, a relative minimum is the least y-value among nearby points on a graph.

TOTD Write (x – 1)(x + 3)(x + 4).

Example 5 Travel: Several popular models of carry-on luggage have a length 10 in. greater than their depth. To comply with airline regulations, the sum of the length, width, and depth, may not exceed 40 in.  Assume that the sum of the length, width, and depth is 40 in. Graph the function relating volume V to depth x. Find the x- intercepts. What do they represent?  

Example 5 Cont Describe a realistic domain. What is the maximum possible volume of a piece of luggage? What are the corresponding dimensions of the luggage?

Example 6 Find the zeros of each function. Then graph the function. 𝒚=(𝒙−𝟐)(𝒙+𝟏)(𝒙+𝟑)

Example 7 Find the zeros of each function. Then graph the function. 𝒚=(𝒙−𝟕)(𝒙−𝟓)(𝒙−𝟑)

Example 8 Write a polynomial function in standard form with the following zeros. −𝟐, 𝟑, 𝟑

Example 9 Write a polynomial function in standard form with the following zeros. −𝟒, −𝟐, 𝟏

Example 10 Write a polynomial function in standard form with the following zeros. −𝟒, −𝟐, 𝟎

Multiplicity Definition 2: If a linear factor of a polynomial is repeated, then the zero is repeated. A repeated zero is called a multiple zero. Definition 3: A multiple zero has a multiplicity equal to the number of times the zero occurs.

Example 11 For each function, find any multiple zeros and state the multiplicity. 𝒇 𝒙 = 𝒙 𝟒 +𝟔 𝒙 𝟑 +𝟖 𝒙 𝟐

Example 12 For each function, find any multiple zeros and state the multiplicity. 𝒇 𝒙 =(𝒙−𝟐)(𝒙+𝟏) 𝒙+𝟏 𝟐

Example 13 For each function, find any multiple zeros and state the multiplicity. 𝒇 𝒙 = 𝒙 𝟑 −𝟒 𝒙 𝟐 +𝟒𝒙

Summary

TOTD Write a polynomial function in standard form with zeros at -5, -4, and 3.