quintic function 4

7/23/2019 Quintic Function 4

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Quintic Function

Polynomials do cover a lot of portion in maths. The highest power of the variables in

a polynomial is termed as its degree. In mathematical language, by quintic function,

one means to refer a polynomial of degree 5.

The general form of a quintic function is given below

f!"# $ a"5 % b"& % c"' % d"( % e" % f ) a* +

here) a,b,c,d,e and f are constant terms, and may belong to the -eld of real

numbers, rational number or comple" numbers. The constant a must not be equal

to /ero, otherwise the polynomial will be of degree & or quartic polynomial.

hen f!"# is set to /ero) provided that a* +, one gets quintic equation as illustrated

below

a"5 % b"& % c"' % d"( % e" % f $ +

GraphThe graph of a quintic function looks quite similar to a cubic function or any odd-degee polynomial.Quintic functions have four local maxima and minima. Since, the quintic equation has positive leadingcoefficient and odd highest power therefore, according to the rule of polynomial graphing, its graph risesto right side and falls to left side.


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The graph of quintic function is shown below:

PropertiesFollowing are the properties that are possessed by quintic function:1)!t can have five roots, since it is a polynomial of degree five. Though, one or more roots may be "erobut a quintic function has at most five roots.

2)There are at most four local extrema in the graph of quintic function. !n other words, this function has"ero to four local maxima or minima, when graphically represented.

3) Quintic function does not characteri"e any general symmetry. !ts graph is asymmetric.

) The highest number of inflection points can be from one to three in quintic function graph.

!)The roots of quintics are quite difficult to solve by radicals. #ut, there are few quintic equations that are

solvable.

How to Solve Quintic Functions?There are few methods by which quintic functions can be solved" These are listed below:

1)$ew quintic equations may be solved with the help of simple factori#ation techniques. Though, allcannot be solved by this method.

2)Quintics may be solved by using $acobi theta function which is an elliptic analog of exponential


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function.

3)Quinitics, sometimes, may be solved using differential equations.

)Tschirnhaus transformationwas introduced by mathematician %hrenfried Tschirnhausenin late%&thcentury. !t is actually a mapping on polynomials and which field theory. This transformation may beused to solve an irreducible quintic with roots to rational function.

!)'ven graphical method may also be used to find the roots of a quintic function or equation.

Examples&ll the quintic equations are not easy to deal with" 'n fact( many of them are not solvable" ere( let

us have a loo* at some examples of simple solvable quintics"

%xamples 1 :$ind the roots of m5- ( ) *.

+olution :m5- ( ) *

m5) (

m)35

m) %.+ ,approx)

%xample 2 :Solve "5+2"4+"3=0

+olution :"5+2"4+"3=0

"3("2+2"+1)=0

"3("+1)2=0

"3=0and("+1)2=0

") *, *, *, -%, -%

%xample 3 :/alculate the roots of following quintic0 "5"4"+1=0

+olution : "5"4"+1=0

"4("1)1("1)=0


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("41)("1)=0

("2+1)("21)("1)=0""""""""" using the identity a2b2=(a+b)(ab)

("2+1)("+1)("1)("1)=0

which gives

("2+1)=00"=1

"1 % ) * 0") - %

"- % ) * 0") %

"- % ) * 0") %

So, ") %, %, %, -%, -%

quintic function 4

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