POLYNOMIALS UNIT Polynomial Basics

A monomial is

A binomial is: A trinomial is:

A polynomial is

Because we write polynomials in descending order: we re-write the function above as:

Term:

Degree:

Coefficient: Leading Coefficient:

Identify all characteristics of the following:

Adding and Subtracting Polynomials:

Like Terms -

(2x3 + 9 – x) + (5x2 + 4 + 7x + x3) (3 – 2x2) – (x2 + 6 – x)

Multiplying and Dividing Polynomials:

3cd2(4c2d – 6cd + 14cd2) (a – 3)(2 – 5a + a2)

(y2 – 7y + 5)(y2 – y – 3)

Pascal's triangle

(k – 5)3 (3x + 1)4

Homework for Polynomial Basics

Find the degree, leading coefficient and number of terms (name of the polynomial).

D:

LC:

Name:

Perform the indicated operation:

POLYNOMIAL UNIT Factoring and Dividing Polynomials

Long Division: (2y2 + 2y3 + 25) ÷ (y – 3)

Synthetic Division: (3x4 – x3 + 5x – 1) ÷ (x + 2)

Synthetic Substitution: P(x) = 2x3 + 5x2 – x + 7 for x = 2

The Remainder Theorem - when dividing by a binomial, the remainder is

The Factor Theorem –

Determine whether the given binomial is a factor of the polynomial P(x).

(x + 2); (3x4 + 6x3 – 5x – 10)

Factoring:

By Grouping: x3 – x2 – 25x + 25. 2x3 + x2 + 8x + 4

Factoring the Difference of Squares: a2 - b2 =

4x2 - 25y2 c2 - 11

The Sum or Difference of Two Cubes:

4x4 + 108x 125d3 – 8

Homework for Factoring and Dividing Polynomials

Divide using Long Division:

Divide using Synthetic Division:

Use Synthetic Substitution to evaluate:

Use the Factor Theorem and Synthetic Division to determine if the binomial is a factor of the

given polynomial:

Factor the following:

POLYNOMIALS UNIT Finding Roots of Polynomial Functions

Find the roots by factoring completely:

The multiplicity of root r is the number of times that x – r is a factor of P(x).

End Behavior: Sketch a graph of the following function based on its factored form. State the end behavior:

Write a polynomial function based on the graph of the following:

Homework for Roots of Polynomial Functions

Sketch a graph of the following polynomial functions and state the end behavior.

Solve the inequality:

Write a polynomial function with the following features:

Write a formula for each polynomial graphed:

,

POLYNOMIAL UNIT Rational Root Theorem and Fundamental Theorem of Algebra

Rational Root Theorem: a way of finding Descartes Rule of Signs: Positive Roots Negative Roots Complex Roots Fundamental Theorem of Algebra: Discuss RRT, Descartes Rule of Signs and FTA with the following: x4 – 3x3 + 5x2 – 27x – 36 = 0 RRT: FTA: Descartes:

A turning point (Critical Point) is where a graph changes from increasing to decreasing or from decreasing to increasing. A turning point corresponds to a local maximum or minimum. Using a Graphics Calculator to find Roots and Max/Mins and Intervals of Increasing/Decreasing. f(x) = 2x3 – 18x + 1

Homework for Rational Root Theorem and Fundamental Thm of Algebra Use a graphics calculator to graph the following and determine: Roots: Max/Mins: Intervals of Increasing/Decreasing: g(x) = x3 – 2x – 3

Use a graphics calculator to graph the following and determine: Roots: Max/Mins: Intervals of Increasing/Decreasing: h(x) = x4 + 4x2 – 6

Use the Rational Root Theorem to find the possible real roots of: 2x3 – 3x2 –10x – 4 = 0 Use the Descartes Rule of Signs to find: # of Positive Real Roots # of Negative Real Roots # of Complex Roots Use the Rational Root Theorem to find the possible real roots of: x3 – 12x2 + 48x – 64 = 0 Use the Descartes Rule of Signs to find: # of Positive Real Roots # of Negative Real Roots # of Complex Roots

POLYNOMIAL UNIT Transforming Polynomial Functions

Let f(x) = x3 + 5x2 – 8x + 1. Write a function g that performs each transformation. Reflect f(x) across the x-axis. Reflect f(x) across the y-axis Let f(x) = 2x4 – 6x2 + 1. Write a new equation for g(x).

Let f(x) = 2x4 – 6x2 + 1. Write a new equation for g(x).

Write a function that transforms f(x) = 6x3 – 3 in each of the following ways. Reflect across the y-axis and shift 2 units down. Write a function that transforms f(x) = 8x3 – 2 in each of the following ways. Compress vertically by a factor of 1/2 , and move the x-intercept 3 units right.

Homework for Transforming Polynomials

For f(x) = x3 + 5, write the rule for g(x) = f(x – 1) – 2 and sketch its graph.

Write a function that reflects f(x) = 2x3 + 1 across the x-axis and shifts it 3 units down.

POLYNOMIAL UNIT

Curve Fitting

The following set of data describes which type of function?

The table shows the closing value of a stock index on the first day of trading for various years. Which type of function best fits the data.

The table below shows the gas consumption of a compact car driven a constant distance at various speed. Write a polynomial function for the data.

Sometimes, it is hard to determine which model best fits the data. In this case, you must look at the regression value (R2). The function that has a regression value The table below shows the opening value of a stock index on the first day of trading in various years. Use a polynomial model to estimate the value on the first day of trading in 1999.

Homework for Curve Fitting