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  Polynomial Functions of Higher Degree

Polynomial Functions of Higher Degree:

Graphs of Polynomials:

The Polynomials are constant and smooth everywhere.

a) A continuous function signifies that it can be drawn devoid of picking up your pencil. There are no holes or jumps in the graph of a polynomial function.

b) The smooth curve signifies that there are no sharp turns (such as an absolute value) in the graph of function.

c) The y-intercept of the polynomial is a constant term a0.

Leading Coefficient Test (right hand behavior):

a) When the leading coefficient, an, of the polynomial is positive, then the right hand side of graph will increase towards + infinity.

b) When the leading coefficient, an, of the polynomial is negative, then the right hand side of graph will drop towards-infinity.

Degree of the Polynomial (left hand behavior):

a) When the degree, n, of the polynomial is even, then the left hand side will do similar as the right hand side.

b) When the degree, n, of the polynomial is odd, then the left hand side will do opposite of the right hand side.

Zeros of a Polynomial Function:

a) The nth degree polynomial in one variable consists of most n real zeros. There are precisely n real or complex zeros.

b) The nth degree polynomial in one variable consists of most n-1 relative extrema (that is, relative maximums or relative minimums). As a relative extremum is a turn in graph, you could as well state there are at most n-1 turns in the graph.

Real Zeros:

When f is a polynomial function in one variable, then the given statements are equivalent.

i) x = a is a zero or root of function f.
ii) x = a is the solution of equation f(x) = 0.
iii) (x-a) is a factor of function f.
iv) (a,0) is the x-intercept of graph of f.

The claim is made that there are maximum real zeros. There is no claim made that they all are unique (that is, different). Some of the roots might be repeated. These are termed as repeated roots. Repeated roots are tied to the concept termed as multiplicity. The multiplicity of a root is a number of times a root is an answer. The simplest way to find out the multiplicity of a root is to look at the exponent on corresponding factor.

Let consider the following:

f(x) = (x-3)^2 (x+5) (x+2)^4

The roots to function will be x = 3 with multiplicity 2, x = -5, and x = -2 with multiplicity 4. It is supposed, and thus unnecessary to write, the multiplicity of 1.

And the beautiful thing is....

The multiplicity of a root, and similarly the exponent on the factor, can be employed to find out the behavior of the graph at zero.

a) When the multiplicity is odd, then the graph will cross the x-axis at zero. That is, it will change the sides or be on opposite sides of x-axis.

b) When the multiplicity is even, then the graph will touch the x-axis at zero. That is, it will stay on similar side of the axis.

Odd Changes, Even Same:

Here are certain places you will be utilizing the concept of Odd Changes, Even stays similar.

A) The left hand behavior of the polynomial function.

  • When the degree of polynomial is Odd, then the left hand Changes from right hand.
  • When the degree of polynomial is Even, then the left hand does Same as right hand.

B) Behavior of the polynomial function at x-intercepts.

  • When the multiplicity is Odd, then the graph will Change sides and cross the axis.
  • When the multiplicity is Even, then the graph will stay on Same side and simply touch the axis.

C) Finding out the solution to inequalities (this is the key in to determining answers really very quickly)

  • When the multiplicity is Odd, the sign will Change at the critical number
  • When the multiplicity is Even, then the sign will remain Same at critical number.

D) Behavior of the rational functions, the x-intercepts.

  • When the multiplicity is Odd, then the graph will Change sides and cross axis.
  • When the multiplicity is Even, the graph will state on Same side and simply touch the axis

E) Vertical asymptotes of the rational functions

  • When the multiplicity is Odd, then the graph will Change sides and one side of vertical asymptote will increase to positive infinity whereas the other side drops to negative infinity.
  • When the multiplicity is Even, then the graph will stay on Same side, and both the sides of vertical asymptote will mount to positive infinity or both the sides will drop to the negative infinity.

F) Finding out the sign of cofactor of an element of the matrix.

  • When the sum of row and column the element is in Odd, then the cofactor will Change and be opposite of the minor.
  • When the sum of column and row the element is in is Even, the cofactor will be the Same as the minor.

Intermediate Value Theorem:

Polynomials are the continuous functions that mean that you cannot pick up your pencil as graphing them.

Question: If at certain point, you are beneath the x-axis, and at the other point you are above the x-axis, and you did not pick up your pencil whereas moving from one point to the other - what happened?

Answer: You crossed x-axis, had a zero or root of the function, found a solution, and so on.

Now, take a concept that a little bit farther. Take any of the two y-values. When they are not similar, then you had to press each and every y-value between the two whenever moving from one to other. The Intermediate Value Theorem defines that formally.

What it is primarily employed for, though, is to determine the zeros of the continuous function.

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