Polynomial Functions: Properties, Equations, and Inequalities, Slides of Elementary Mathematics

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Unit 3

Polynomial Functions

Friday, January 31, 2020

Polynomial Function

A polynomial function is define by a

polynomial expression in x given in the

standard form;

y = a

n

x

n

+ a

n- 1

x

n- 1

+ … +a

1

x + a

0 The value of n must be a positive integer a n- 1 , ..., a 1 , a 0 are called coefficients and these are real numbers

Leading Coefficient

The leading coefficient is the

coefficient of the first term in a

polynomial when the terms are written

in descending order by degrees.

For example, the quartic function

f(x) = - 2x

4

+ x

3

• 5x 2 - 10 has a leading

coefficient of - 2.

Graphs of Polynomial Function Exploring the graphs of polynomial function Linear Function Quadratic Function Cubic Function Quartic Function -10 -8 -6 -4 -2 2 4 6 8 10

• • 2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10

2 4 6 8 10 -10 -8 -6 -4 -2 2 4 6 8 10

2 4 6 8 10 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

5 10

Cubic Polynomials

Equation Factored form & Standard form X- Intercepts Sign of Leading Coefficient End Behaviour Domain and Range Factored y=(x+1)(x+4)(x-2) Standard y=x^3 +3x^2 - 6x- 8

• 4, - 1, 2 Positive As x , y  and x -, y - Domain {x| x Є R } Range {y| y Є R } Factored y=-(x+1)(x+4)(x-2) Standard y=-x^3 - 3x^2 +6x+
• 4, - 1, 2 Negativ e As x , y - and x -, y  Domain {x| x Є R } Range {y| y Є R } The following chart shows the properties of the graphs on the left. -5 -4 -3 -2 -1 1 2 3 4 5

2 4 6 8 10 12 -5 -4 -3 -2 -1 1 2 3 4 5

2 4 6 8 10 12

Quartic Polynomials

Look at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?
2. What is the end behaviour for each graph?
3. Which graph do you think has a positive leading coeffient? Why?
   5. Which graph do you think has a negative leading coefficient? Why?
4. How many x-intercepts do graphs A & B have? Graph A Graph B -5 -4 -3 -2 -1 1 2 3 4 5 - - - - - 2 4 6 8 10 12 14 -5 -4 -3 -2 -1 1 2 3 4 5

2 4 6 8 10

Polynomial Equation

If y = f(x) is a polynomial function then

f(x) = 0 is a polynomial equation.

Example 1.) F(x) = 2x 4 ‒ 3x 3

• x ‒ 4 2.) F(x) = 4x 4
• 3x 3
• 2x + 1 2x 4 ‒ 3x 3
• x ‒ 4 = 0 4x 4
• 3x 3
• 2x + 1 = 0 polynomial function polynomial equation Note: The solution to a polynomial equation is called the zero of the polynomial function.

Remainder Theorem

If a polynomial function P(x) is divided by x – a, then the remainder is number P(a) where the function is evaluated at x = a 1.) F(x) = 2x 4 ‒ 3x 3

• x ‒ 4 divided by (x – 2) 3.) F(x) = 2x³ + 1 ‒ x + 3x² divided by (x + 2) 4.) F(x) = 2x³ ‒ 3x² ‒ x + 2 divided by (x – 1) 2.) F(x) = 4x 4
• 3x 3
• 2x + 1 divided by (x + 1) Find the remainder of each R = 6 R = 0 R = - 1 R = 0

Fundamental Theorem of Algebra

Every polynomial function of degree n≥

has at least one complex zero.

Every polynomial function of degree n≥

has exactly n complex zeros counting

multiplicities.

Example:

1.) F(x) = 2x 4 ‒ 3x 3

• x ‒ 4 Degree 4 has exactly 4 complex zeros or solution 2.) F(x) = 2x³ + 1 ‒ x + 3x² (^) Degree 3 has exactly 3 complex zeros or solution

Solving Polynomial Equation

• Solving Polynomial Equation in

Factored Form

• Example:

1.) (x – 2)(2x – 1)(x + 1) = 0

x – 2 = 0 x = 2 2x – 1 = 0 2x = 1 (^2 ) x =

x + 1 = 0 x = - 1

Finding Polynomial Function or Equation

Finding a Polynomial Function or Equation

given its Roots, Zeros or Solution.

Example:

1. Find a Polynomial function whose

zeros are x = 1 and x = - 3

Solution:

f(x) (^) = (x – 1) (x + 3) f(x) (^) **= x 2

• 3x -**^ 1x^ -^3 f(x) (^) **= x 2
• 2x - 3 or**^ **x 2
• 2x – 3 = 0**

Finding Polynomial Function or Equation

Example:

2. Find a Polynomial function whose

zeros are x = - 2, x = - 1 and x = 2

Solution:

f(x) (^) = (x + 2) (^) (x + 1) f(x) (^) **= (x 2

• 1x + 2x**^ **+ 2) f(x) = (x 2
• 3x + 2) (x – 2) (x - 2) (x - 2) = x 3**

- 2x 2 + 3x 2 - 6x (^) + 2x (^) **- 4 = x 3

• 1x** (^2) **- 4x

• 4 f(x) f(x)**

Solving Polynomial Equation

Solving for the solutions or zeroes

of a polynomial equation or function

can be done by:

1. Graphing calculator, Synthetic

Division and Factoring.

2. Rational Root Theorem,

Synthetic Division and Factoring.

Solving Polynomial Equation

Solving by Graphing Calculator,

Synthetic Division and Factoring

Example: Find the zeroes or solution

1. f(x) = 3x

3

+ x

2

• 3x – 1 Step 1: Use graphing calculator to get as many exact zero as possible by doing y = 3x 3

• x 2

• 3x - 1 and do 2nd TRACE ZERO. Step 2: Use synthetic Division to reduced the function to a lower degree.