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POLYNOMIAL FUNCTIONS

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Fairooz Anwar

on 10 January 2014

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Transcript of POLYNOMIAL FUNCTIONS

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
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