If x 2 is a polynomial 2

content
" Preliminary remarks
" The formula
“Geometric analysis
“Zeros and other points
“Examples

Preliminary remarks

This chapter is about the polynomial functions. The two most important polynomial functions, the linear function and the quadratic polynomial can also be found here.

You can also read all the power functions with natural exponents here soon.

As we will analyze geometrically, certain patterns repeat themselves again and again in polynomial functions, which is why we will keep our formulas general but focus primarily on third and fourth degree polynomials in examples.

The formula

A polynomial function of degree \ (n \) is a function of form
\ begin {align *}
f (x) = a_0 + a_1x + a_2x ^ 2 + \ cdots a_nx ^ n
\ end {align *}
The parameters \ (a_0 \) to \ (a_ {n-1} \) are from \ (\ mathbb {R} \). It is assumed \ (a_n \ in \ mathbb {R} \ setminus \ {0 \} \), because if \ (a_n = 0 \) our polynomial would only have degree \ (n-1 \).

Let us consider our examples from the introduction, for which the line \ (f \) was true
\ begin {align *}
f (x) = - \ frac {2} {3} x + 5,
\ end {align *}
In the notation of the linear functions we denoted \ (5 = d \) as the y-section and \ (- \ frac {2} {3} = k \) as the slope. Considered as a polynomial, \ (a_0 = 5 \) and \ (a_1 = - \ frac {2} {3} \) and the linear function is a first order polynomial (\ (x = x ^ 1 \)). In the case of quadratic functions such as \ (g \), we denoted the parameters by \ (a, b, c \) due to the similarity to the solution formula. As a polynomial function we get
\ begin {align *}
& g (x) = \ frac {1} {4} (x + 1) ^ 2 + 1 = \ frac {1} {4} x ^ 2 + \ frac {1} {2} x + \ frac {5} {4} \
& \ Rightarrow a_0 = \ frac {5} {4}, a_1 = \ frac {1} {2}, a_2 = \ frac {1} {4}
\ end {align *}


and the quadratic function is a second degree polynomial.

In our power functions \ (h \) and \ (p \) with
\ begin {align *}
& h (x) = \ frac {1} {3} x ^ 3 + 1 \
& p (x) = - \ frac {1} {3} x ^ 4 + 4
\ end {align *}


we get at \ (h \) for \ (a_3 = \ frac {1} {3} \), \ (a_0 = 1 \) and \ (a_1 = a_2 = 0 \). Similarly, we have \ (p \) \ (a_4 = - \ frac {1} {3} \), \ (a_0 = 4 \) and \ (a_1 = a_2 = a_3 = 0 \). It is often said that "the middle links are zero". Then it holds that a polynomial function of degree \ (n \) is a power function if \ (a_ {n-1} = \ dots = a_1 = 0 \) holds.

Geometric analysis

If we look at the graphs of all four polynomial functions, we see that they all pass through \ ((0; a_0) \),

the constant component of a polynomial function indicates the \ (y \) section not only for the linear function but also for every polynomial, since \ (f \) applies to any polynomial
\ begin {align *}
f (0) = a_0 + a_1 \ cdot 0 + a_2 \ cdot 0 ^ 2 + \ cdots a_n \ cdot 0 ^ n = a_0
\ end {align *}
from which the point \ ((0; a_0) \) results. Let us now consider polynomials whose middle terms are not 0 and start with polynomials of degree \ (n = 3 \), we consider
\ begin {align *}
& f (x) = - \ frac {1} {8} x ^ 3- \ frac {1} {8} x + 4 \
& g (x) = \ frac {1} {6} x ^ 3- \ frac {1} {4} x ^ 2-3x \
& p (x) = x ^ 3 + 18x ^ 2 + 108x + 220
\ end {align *}


and see that, similar to the power functions of degree 3, the factor \ (a_3 \) before \ (x ^ 3 \) decides whether the polynomial "runs upwards or downwards" for any polynomial functions of the third degree. The polynomial function \ (f \) "runs primarily" downwards due to \ (a_3 = \ frac {-1} {8} \), the other two upwards. "New" compared to power functions of degree 3, however, are the "waves", the monotonic change, in the marked area. Is that due to the middle links of \ (g \)? However, the functions \ (f \) and \ (p \) also have middle terms other than 0 and are similar to our known power function. A complete analysis of this problem is only possible with the help of differential calculus, but it can be shown that there are three forms of third-degree polynomials which, depending on the sign of \ (a_3 \), go up

or down

watch. Similar observations can be made for the fourth degree polynomials. There are three "different forms" and depending on \ (a_4 \) in \ (a_4x ^ 4 \) the polynomial functions look up

 

or down

 

Except with a table of values, it is therefore difficult for us to sketch any polynomial function. However, the following points often help us.

Zeros and other points

The zeros of a polynomial \ (f \) correspond to the solutions of the equation
\ begin {align *}
a_0 + a_1x + a_2x ^ 2 + \ cdots a_nx ^ n = 0.
\ end {align *}
According to Gauss's theorem and its conclusions

  • this equation has at most \ (n \) solutions in \ (\ mathbb {R} \).
  • this equation has at least one solution in \ (\ mathbb {R} \) if \ (n \) is odd.

This coincides with our previous knowledge that a linear function, a polynomial of the first degree always has a zero and a quadratic function, a polynomial of the second degree, has 0.1 or 2 zeros. We now know that a third-degree polynomial has at least one and a maximum of three zeros, this is in line with our geometrical considerations before. In general, we can no longer solve higher-order equations by hand. Possibilities to sometimes find the zeros anyway are guessing zeros, polynomial division, factoring and substitution.

For local extrema, i.e. minima and maxima, it applies that a polynomial function of degree \ (n \)

  • has a maximum of \ (n-1 \) extrema.
  • has at least one extreme if \ (n \) is even.

Colloquially, a polynomial with \ (n \) odd has to go through the \ (x \) axis from top to bottom at some point. So it has at least one zero. Similarly, if a polynomial with \ (n \) comes from above, it has to go up again at some point, so it needs a minimum.

Examples

Analysis of graphs: Analyze the following graph of the function \ (g \) and read off as much information as possible.

solution

From the basic form we can see that \ (n \) is straight. In addition, \ (a_n> 0 \) applies, because the function looks up. We have three extremes, so the degree is at least four!

Annotation: The degree of the polynomial function is actually four. However, we find graphs with, for example, degree 6, which also fulfill the same properties.

Find parameters: The polynomial function \ (g (x) = \ frac {x ^ 4} {5} - \ frac {x ^ 3} {5} - \ frac {4 x ^ 2} {5} + \ frac {4 x is given } {5} + a_0 \) with graph

solution

We know that every polynomial function goes through the point \ ((0; a_0) \). The graph of \ (g \) runs through \ ((0; 0) \), so because

\ (g (0) = \ frac {0} {5} - \ frac {0} {5} - \ frac {0} {5} + \ frac {0} {5} + a_0 = 0 \)

then \ (a_0 = 0 \) hold.

Find zeros: The function \ (g (x) = \ frac {x ^ 4} {5} - \ frac {x ^ 3} {5} - \ frac {4 x ^ 2} {5} + \ frac {4 x is given } {5} \). Find all zeros.

solution

We forget, of course, that we have just seen the graph of \ (g \). We have to use the equation to find the zeros
\ begin {align *}
& g (x) = 0 \
& \ frac {x ^ 4} {5} - \ frac {x ^ 3} {5} - \ frac {4 x ^ 2} {5} + \ frac {4 x} {5} = 0
\ end {align *}
solve and receive with a few skillful steps
\ begin {align *}
& \ frac {x ^ 4} {5} - \ frac {x ^ 3} {5} - \ frac {4 x ^ 2} {5} + \ frac {4 x} {5} = 0 \
& x ^ 4-x ^ 3-4x ^ 2 + 4x = 0 \
& x \ cdot (x ^ 3-x ^ 2-4x + 4) = 0.
\ end {align *}


Due to the product theorem, we therefore get the first zero \ (x_1 = 0 \) and only need the equation
\ begin {align *}
x ^ 3-x ^ 2-4x + 4 = 0
\ end {align *}
to solve. In the chapter Zeros guess LINK we learn that we assume that the other zeros are from \ (\ {\ pm 1, \ pm 2, \ pm 4 \} \), since these are the divisors of 4. We try \ (x = 1 \) and actually get \ (1 ^ 3-1 ^ 2-4 \ cdot 1 + 4 = 0 \). Our second zero is at the point \ (x_2 = 1 \). The polynomial division with \ ((x-1) \) then brings us
\ begin {align *}
x ^ 3-x ^ 2-4x + 4 = (x-1) (x ^ 2-4)
\ end {align *}
whence it results that only the equation
\ begin {align *}
x ^ 2-4 = 0
\ end {align *}
is to be solved. We get \ (x_3 = 2 \) and \ (x_4 = -2 \) as solutions.

Annotation: With the help of the four zeros we can already sketch the function. Assume \ (g (x) = \ frac {x ^ 4} {5} - \ frac {x ^ 3} {5} - \ frac {4 x ^ 2} {5} + \ frac {4 x} { 5} \) is given and we have calculated our zeros. Then we can draw them in and with our knowledge of the approximate shape of a fourth degree polynomial function draw them in.

The exact course is unknown to us except for the zeros, which is why comparable polynomial functions have been drawn in.