C. Higher Functions Pre-Calculus 30

PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

Key Terms

1. Characteristics of Polynomial Functions PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

1. Characteristics of Polynomial Functions

What is a Polynomial Function?

A polynomial function has the form Where n = is a whole number x = is a variable The coefficients an to a0 are real numbers The degree of the poly function is n, the exponent of the greatest power of x The leading coefficient is an the coefficient of the greatest power of x The constant term is a0

Example 1

Types of Polynomial Functions:

Each graph has at least one less change of direction then the degree of its function.

Characteristics of Polynomial Functions:

Example 2

Example 3

Key Ideas p.113

Practice Ex. 3.1 (p.114) #1-4 odds in each, 5-10 # 1-4 odds in each, 5-13 odds

2. Finding X-Intercepts PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

2. Finding X-Intercepts When only given the equation of a function what are some strategies that we have use to find x-intercepts? Sub in a zero and solve Factor Quadratic Formula Decompostion

Which of these strategies will work if our polynomial function have a degree greater than 2?

In this section we will look at some methods to completely factor a polynomial function with a degree greater than 2 in order to find the zeros (x-intercepts).

Long Division Long division of a polynomial is done just like your do with numbers but now you have variables

You use long division to divide a polynomial by a binomial: The dividend, P(x), which is the polynomial that is being divided The divisor, x-a, which is the binomial being divided into the polynomial The quotient, Q(x), which is the expression that results from the division The remainder, R, which is what is left over when the division is done.

To Check the division of the polynomial, verify the statement 𝑃 𝑥 = 𝑥−𝑎 𝑄 𝑥 +𝑅

Example 1  

Example 2

If we graphed the polynomial function from example 2 what would the x-intercepts be?

Long division gives is factors of the polynomial function which when set equal to zero and solved are x-intercepts, zeros, or roots.

Synthetic Division: A short form of division that uses only the coefficients of the terms It involves fewer calculations

Example 3

Remainder Theorem: When a polynomial P(x) is divided by a binomial x-a, the remainder is P(a) If the remainder is 0 then the binomial x-a is a factor of P(x) If the remainder is not 0 then the binomial x-a is NOT a factor of P(x)

Example 4

Key Ideas p.123

Practice Ex. 3.2 (p.124) #1,2,3-7 odds in each, 8-13 #2, 3-7 odds in each, 8, 9-17 odds

3. Factoring Completely PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

3. Factoring Completely Last day we looked at how dividing polynomial functions by a binomial shows us if that binomial is a factor or not We also discussed how we want our equations in factored form because that gives is the zeros/roots/x-intercepts Today we are going to extend that idea

Factor Theorem: The factor theorem states that x-a is a factor of a polynomial P(x) if and only if P(a)=0 If and only if means that the result works both ways. That is, If x-a is a factor then P(a)=0 If P(a)=0, then x-a is a factor of a polynomial P(x)

For example, which binomial are factors of the polynomial 𝑃 𝑥 = 𝑥 3 +4 𝑥 2 +𝑥−6?

The zeros of a polynomial function are directly related to the x-intercepts If we graph 𝑃 𝑥 = 𝑥 3 +4 𝑥 2 +𝑥−6 we will see the zero/x-intercepts are at x=1, x= -2, and x= -3 The corresponding factors are x-1, x+2, x+3 So to find the x-intercepts given the polynomial functions equation we have to get the equation in fully factored form.

Example 1

When factoring a polynomial function sometimes the most difficult part is deciding which values of “a” we should use when using long division, synthetic division or factor theorem.

Consider the Polynomial function 𝑃 𝑥 = 𝑥 3 −7 𝑥 2 +14𝑥−8 We want P(x)=0 when x=a so……..

This is referred to as the Integral Zero Theorem The integral zero theorem describes the relationship between the factors and the constant term of a polynomial. The theorem states that if x-a is a factor of a polynomial P(x) with integral coefficients, then “a” is a factor of the constant term of P(x) and x=a is a integral zero of P(x).

Example 2

Factor by Grouping: If a polynomial P(x) has an even number of terms, it may be possible to group tow terms at a time and remove a common factor If the binomial that results from common factoring is the same for each pair of terms, then P(x) may be factored by grouping Will not always work!!!!

Steps to factoring Polynomial Functions: 1. Use the integral zero theorem to list possible integer values for zeros 2. You can use the factor theorem to determine if the values that are zeros (this take a lot of time so I don’t suggest it) 2. Use one type of division to determine all the factors 3. Write equation in factored form

Example 3

Example 4

Key Ideas p. 133

Practice Ex. 3.3 (p.133) #1-7 odds in each, 8-14 evens

4. Graphing Polynomial Functions PC30.10 Demonstrate understanding of polynomials and polynomial functions of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients).

4. Graphing Polynomial Functions We are now going to use all the information we have learned in this unit to this point along with a little new into to Graph Polynomial Functions without a graphing calculator

For example: 𝑓 𝑥 =(𝑥−1)(𝑥+2)(𝑥+2) (𝑥−5) 3

Example 1

Example 2

We can also graph polynomial functions using transformations is given the function in the form 𝑦=𝑎 [𝑏 𝑥−ℎ ] 𝑛 +𝑘

Example 3

Example 4

Key Ideas p. 147

Practice Ex. 3.4 (p.147) #1-10 odds in questions with multiple parts, 12-18 evens #3-10 odds in questions with multiple parts, 11-23 odds