**POLYNOMIAL FUNCTIONS**

The expression contains only one variable, with the powers arranged in descending order.

Pol·y·no·mi·al

EXPLORING POLYNOMIAL FUNCTIONS

CHARACTERISTICS OF POLYNOMIAL FUNCTIONS

Polynomial functions

End Behaviours

CHARACTERISTICS OF POLYNOMIAL FUNCTIONS IN FACTORED FORM

The zeros of the polynomial function f(x) are the same as the roots.

If any of the factors are squared, then the corresponding x-intercepts are turning points of the curve and the x-axis is tangent to the curve at these points.

If any of the factors are cubed, then the corresponding x-intercepts are points where the x-axis is tangent to the curve and also passes through the x-axis.

Family of polynomial functions

TRANSFORMATIONS OF CUBIC AND QUARTIC FUNCTIONS

Example 1

To sketch the graph of

vertically stretch the graph of by a factor of 2,

reflect it through the x-axis,

translate it 3 units to the right and 5 units up.

As a result of these transformations, every point on the graph of changes to

Example 2

DIVIDING POLYNOMIALS

Ways to divide polynomials

Long Division

Synthetic Division

FACTORING POLYNOMIALS

Factor theorem

a special case that helps make sense between a polynomial and its remainder when divided by a polynomial.

Example 1

FACTORING A SUM OF DIFFERENCE OF CUBES

SUM OF CUBES

An expression that contains two perfect cubes that are added together

DIFFERENCE OF CUBES

An expression that contains perfect cubes where one is subtracted from the other

Synthetic Division cont'd

Example

Write the expression as

Since this is a difference of cube

Simplify to

to get factors if A=x and B=4

Example

Since this is a sum of cube

Write the expression as

Simplify to

to get factors, if A=x and B=1

Remainder theorem

When a polynomial, f(x), is divided by x-a, the remainder is equal to f(a).

Example 2

First, divide out any common

factors of the polynomial.

Use the factor theorem to factor

the remaining cubic.

Divide to determine the other

factors.

Factor the quotient.

Factor the quotient.

Divide again to determine the other

factors.

Divide to determine the other

factors.

Use the factor theorem to factor the cubic.

Use the factor theorem again to factor

the remaining cubic.

Write the terms of the dividend and the quotient in descending order, by degree.

Follow the same steps that you use for long division with numbers.

Repeat this process until the degree of the remainder is less than the degree of the divisor.

The remainder is -87

Given

x-3=0 therefore k=3

Create a chart that contains the

coefficients of the dividend, as shown.

The dividend and binomial must be

written with its terms in descending

order, by degree.

Bring the first term down. This is now

the coefficient of the first term of the

quotient.

Multiply it by k, and write the answer

below the second term of the

dividend.

Now add the terms together.

Repeat this process for the answer

you just obtained.

Repeat this process one last time.

The last number below the chart is

the remainder. The first numbers are

the coefficients of the quotient,

starting with the degree that is one

less than the original dividend.

When this is graphed...

a= 1

k= 1

d= -9

c= 0

Therefore

Definitions

the coefficient of the term with the highest degree in a polynomial

the greatest/least value attained by a function for all values in its domain

An odd-degree polynomial function has opposite end behaviours

If the leading coefficient is negative, then the function extends from the second quadrant to the fourth quadrant. This means that as

If the leading coefficient is positive, then the function extends from the third quadrant to the first quadrant. This means that as

An even-degree polynomial function has the same end behaviours

If the leading coefficient is positive, then the function extends from the second quadrant to the first quadrant. This means that as

If the leading coefficient is negative, then the function extends from the third quadrant to the fourth quadrant. This means that as

Turning points

A polynomial function of degree n has at most n-1 turning points.

For example:

Number of Zeroes

A polynomial function of even degree may have no zeros.

Symmetry

Most polynomial functions have no symmetrical properties so they are neither even nor odd, with no relationship between f(-x) and f(x).

Since the function has a degree of 4, it will have at least 1 turning point and at most 4-1 or 3 turning points.

Can be factored into this:

Can be factored into this:

Theorems

x-a is a factor of f(x), only if f(a)=0

Factor a polynomial of degree 3 or greater

use the Factor Theorem to determine a factor of f(x)

divide f(x) by x-a

factor the quotient, if possible

may be necessary to use the factor theorem more than once

Not all polynomial functions can be factored

by a polynomial of the same degree or less using long division.

Synthetic division is a shorter form of polynomial division.

can only be used when the divisor is linear that is, x-d or x+d

Steps

terms should be arranged in descending order of degree, in both the divisor and the dividend

zero must be used as the coefficient of any missing powers of the variable in both the divisor and the dividend

If the remainder is zero, then both thedivisor and the quotient are factors of the dividend.

a: vertical stretch/compression and possibly a vertical reflection

k: horizontal stretch/compression and possibly a horizontal reflection

d: horizontal translation

c: a vertical translation

can be graphed by applying transformations to the graph of the parent function

where

Each point (x,y) on the graph of the parent function changes to

a set of polynomial functions whose equations have the same degree and whose graphs have common characteristics

Factoring

Substitute the zeros into the general equation of the

appropriate family of polynomial functions

Substitute the coordinates of an additional point for x and y, and solve for a to determine the equation

Linear polynomial functions

(order 1)

If any of the factors are linear, corresponding x-intercept is a point where the curve passes through the x-axis

graph has a linear shape near this x-intercept

Squared polynomial functions

(order 2)

The graph has a parabolic shape near these x-intercepts.

Cubed polynomial functions

(order 3)

The graph has a cubic shape near these x-intercepts.

Absolute maximum/ absolute minimum

Leading coefficient

Polynomial functions of the same degree have similar characteristics.

The degree and the leading coefficient in the equation indicate the end behaviours of the graph.

The degree provides information about the shape, turning points, and zeros of the graph.

A polynomial function of degree n may have up to n distinct zeros.

A polynomial function of odd degree must have at least one zero.

Some polynomial functions are symmetrical in the y-axis. These are even functions, where f(-x)= f(x).

Some polynomial functions have rotational symmetry about the origin. These are odd functions, where f(x)= -f(x).

An expression of the form

where

are real numbers and n is a whole number.

the exponents on the variable must be whole

numbers

The domain and the range of a polynomial function is all real numbers, but range may have a lower bound or an upper bound (but not both).

The graphs do not have horizontal or vertical asymptotes

Examples