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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:

$f (x)$ = $a x^{5}$ + $b x^{4}$ + $c x^{3}$ + $d x^{2}$ + $e x$ + $f$ ; $a \neq$ 0

Where; $a, b, c, d, e$ and $f$ are constant terms, and may belong to the field of real numbers, rational number or complex numbers. The constant "a" must not be equal to zero, otherwise the polynomial will be of degree 4 or quartic polynomial.
When $f (x)$ is set to zero; provided that $a \neq$ 0, one gets quintic equation as illustrated below:
$a x^{5}$ + $b x^{4}$ + $c x^{3}$ + $d x^{2}$ + $e x$ + $f$ = 0

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Graph

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The 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 leading coefficient and odd highest power ; therefore, according to the rule of polynomial graphing, its graph rises to right side and falls to left side.
The graph of quintic function is shown below:
quintic Function Graph

Properties

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Following are the properties that are possessed by quintic function:
1) It can have five roots, since it is a polynomial of degree five. Though, one or more roots may be zero; but a quintic function has at most five roots.

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

3) Quintic function does not characterize any general symmetry. Its graph is asymmetric.

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

5) The roots of quintics are quite difficult to solve by radicals. But, there are few quintic equations that are solvable.

How to Solve Quintic Functions?

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There are few methods by which quintic functions can be solved. These are listed below:

1) Few quintic equations may be solved with the help of simple factorization techniques. Though, all cannot be solved by this method.

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

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

4) Tschirnhaus transformation was introduced by mathematician Ehrenfried Tschirnhausen in late 16$^{th}$ century. It is actually a mapping on polynomials and which field theory. This transformation may be used to solve an irreducible quintic with roots to rational function.

5) Even graphical method may also be used to find the roots of a quintic function or equation.

Examples

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All the quintic equations are not easy to deal with. In fact, many of them are not solvable. Here, let us have a look at some examples of simple solvable quintics.

Examples 1 : Find the roots of $m^{5}$ - 3 = 0.

Solution : $m^{5}$ - 3 = 0

$m^{5}$ = 3

$m$ = $\sqrt[5]{3}$

$m$ = 1.2457 (approx)

Example 2 : Solve $x^{5}+2x^{4}+x^{3} = 0$

Solution : $x^{5}+2x^{4}+x^{3} = 0$

$x^{3}(x^{2}+2x+1) = 0$

$x^{3}(x+1)^{2} = 0$

$x^{3} = 0$ and $(x+1)^{2}=0$

$x$ = 0, 0, 0, -1, -1

Example 3 : Calculate the roots of following quintic: $x^{5}-x^{4}-x+1=0$

Solution : $x^{5}-x^{4}-x+1=0$

$x^{4}(x-1)-1(x-1)=0$

$(x^{4}-1)(x-1)=0$

$(x^{2}+1)(x^{2}-1)(x-1)=0$ ......... using the identity $a^{2}-b^{2}=(a+b)(a-b)$

$(x^{2}+1)(x+1)(x-1)(x-1)=0$

which gives

$(x^{2}+1)=0 \Rightarrow x = \pm 1$

$x$ + 1 = 0 $\Rightarrow$ $x$ = - 1

$x$ - 1 = 0 $\Rightarrow$ $x$ = 1

$x$ - 1 = 0 $\Rightarrow$ $x$ = 1

So, $x$ = 1, 1, 1, -1, -1
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