Fourth degree polynomial

In algebra , a fourth degree polynomial is a polynomial of the form

{\ displaystyle f (x) = ax ^ {4} + bx ^ {3} + cx ^ {2} + dx + e,}

with non-zero. A quartic function is the mapping corresponding to this polynomial . ${\ displaystyle a}$ ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$

A quartic or fourth degree equation is an equation of form

{\ displaystyle ax ^ {4} + bx ^ {3} + cx ^ {2} + dx + e = 0,}

with . ${\ displaystyle a \ not = 0}$

Properties of quartic functions

In the following is a quartic function defined by with . ${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$ ${\ displaystyle f (x) = ax ^ {4} + bx ^ {3} + cx ^ {2} + dx + e,}$ ${\ displaystyle a \ not = 0}$

Behavior in infinity

As with all rational functions of an even degree, the following applies

{\ displaystyle \ lim \ limits _ {x \ to + \ infty} f (x) = + \ infty}

, ,

{\ displaystyle \ lim \ limits _ {x \ to - \ infty} f (x) = + \ infty}

if the leading coefficient is positive, and ${\ displaystyle a}$

{\ displaystyle \ lim \ limits _ {x \ to + \ infty} f (x) = - \ infty}

, ,

{\ displaystyle \ lim \ limits _ {x \ to - \ infty} f (x) = - \ infty}

if is negative. ${\ displaystyle a}$

zeropoint

A fourth degree polynomial has a maximum of four zeros, but it cannot have real zeros either. If zeros are counted according to their multiplicity , it has exactly four complex zeros. If all zeros are real, the discriminant is nonnegative. The converse does not apply, the polynomial has positive discriminants but no real zeros. ${\ displaystyle x ^ {4} +4}$

There is a solution formula for the (complex) zeros, see quartic equation . The numerical finding of real zeros is possible, for example, with the Newton method .

Local extremes

As a polynomial function, it can be differentiated as often as desired ; the cubic function results for its 1st derivative ${\ displaystyle f}$ ${\ displaystyle f '}$

{\ displaystyle f '(x) = 4ax ^ {3} + 3bx ^ {2} + 2cx + d}

.

If its discriminant is positive, then it has exactly three local extrema, namely for a local maximum and two local minima or for two local maxima and a local minimum. ${\ displaystyle f}$ ${\ displaystyle a> 0}$ ${\ displaystyle a <0}$

Turning points

A quartic function has at most two turning points . The turning points are the zeros of the 2nd derivative . ${\ displaystyle f}$ ${\ displaystyle (x_ {W}; f (x_ {W}))}$ ${\ displaystyle x_ {W}}$ ${\ displaystyle f '' (x) = 12ax ^ {2} + 6bx + 2c}$

Fourth degree polynomials

Be any ring . Fourth degree over polynomials are expressions of the form ${\ displaystyle R}$ ${\ displaystyle R}$

{\ displaystyle ax ^ {4} + bx ^ {3} + cx ^ {2} + dx + e \ in R [x]}

with and . Formally, these are elements of the polynomial ring of degree 4, they define mappings from to . For are quartic functions in the above sense. ${\ displaystyle a, b, c, d, e \ in R}$ ${\ displaystyle a \ not = 0}$ ${\ displaystyle R}$ ${\ displaystyle R}$ ${\ displaystyle R = \ mathbb {R}}$

If is an algebraically closed field , every polynomial of the fourth degree decays as the product of four linear factors. ${\ displaystyle R}$

More generally, quartic polynomials in variables are expressions of the form ${\ displaystyle n}$

{\ displaystyle \ sum _ {i, j, k, l = 1} ^ {n} a_ {i, j, k, l} x_ {i} x_ {j} x_ {k} x_ {l} + \ sum _ {i, j, k = 1} ^ {n} b_ {i, j, k} x_ {i} x_ {j} x_ {k} + \ sum _ {i, j = 1} ^ {n} c_ {i, j} x_ {i} x_ {j} + \ sum _ {i = 1} ^ {n} d_ {i} x_ {i} + e \ in R [x_ {1}, \ ldots, x_ { n}]}

,

whereby not all should be zero. These polynomials define mappings from to . Their sets of zeros im are referred to as quartic curves and as quartic areas . ${\ displaystyle a_ {i, j, k, l}}$ ${\ displaystyle R ^ {n}}$ ${\ displaystyle R}$ ${\ displaystyle R ^ {n}}$ ${\ displaystyle n = 2}$ ${\ displaystyle n = 3}$

Solution of the fourth degree equation by radicals (root expressions)

Nature of the solutions

For the quartic equation

{\ displaystyle ax ^ {4} + bx ^ {3} + cx ^ {2} + dx + e = 0}

with real coefficients and the nature of the roots (the solutions) is essentially given by the sign of the so-called discriminant ${\ displaystyle a, b, c, d, e}$ ${\ displaystyle a \ neq 0}$

{\ displaystyle {\ begin {aligned} \ Delta \ = \ & 256a ^ {3} e ^ {3} -192a ^ {2} bde ^ {2} -128a ^ {2} c ^ {2} e ^ {2 } + 144a ^ {2} cd ^ {2} e-27a ^ {2} d ^ {4} \\ & + 144ab ^ {2} ce ^ {2} -6ab ^ {2} d ^ {2} e -80abc ^ {2} de + 18abcd ^ {3} + 16ac ^ {4} e \\ & - 4ac ^ {3} d ^ {2} -27b ^ {4} e ^ {2} + 18b ^ {3 } cde-4b ^ {3} d ^ {3} -4b ^ {2} c ^ {3} e + b ^ {2} c ^ {2} d ^ {2} \ end {aligned}}}

In addition, you have to consider four more polynomials. This gives information about how many zeros are real and how many are really complex.

General formulas for the roots

Full solution formula. Too complicated to be really useful.

The four roots , , and of the general quartic equation ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle x_ {3}}$ ${\ displaystyle x_ {4}}$

{\ displaystyle ax ^ {4} + bx ^ {3} + cx ^ {2} + dx + e = 0 \,}

with a ≠ 0 result from the following formula.

{\ displaystyle {\ begin {aligned} x_ {1,2} \ & = - {\ frac {b} {4a}} - S \ pm {\ frac {1} {2}} {\ sqrt {-4S ^ {2} -2p + {\ frac {q} {S}}}} \\ x_ {3,4} \ & = - {\ frac {b} {4a}} + S \ pm {\ frac {1} { 2}} {\ sqrt {-4S ^ {2} -2p - {\ frac {q} {S}}}} \ end {aligned}}}

with p and q as follows

{\ displaystyle {\ begin {aligned} p & = {\ frac {8ac-3b ^ {2}} {8a ^ {2}}} \\ q & = {\ frac {b ^ {3} -4abc + 8a ^ { 2} d} {8a ^ {3}}} \ end {aligned}}}

in which

{\ displaystyle {\ begin {aligned} S & = {\ frac {1} {2}} {\ sqrt {- {\ frac {2} {3}} \ p + {\ frac {1} {3a}} \ left (Q + {\ frac {\ Delta _ {0}} {Q}} \ right)}} \\ Q & = {\ sqrt [{3}] {\ frac {\ Delta _ {1} + {\ sqrt {\ Delta _ {1} ^ {2} -4 \ Delta _ {0} ^ {3}}}} {2}}} \ end {aligned}}}

(if or , see under special cases of the formula below) ${\ displaystyle S {=} 0}$ ${\ displaystyle Q {=} 0}$

here is

{\ displaystyle {\ begin {aligned} \ Delta _ {0} & = c ^ {2} -3bd + 12ae \\\ Delta _ {1} & = 2c ^ {3} -9bcd + 27b ^ {2} e + 27ad ^ {2} -72ace \ end {aligned}}}

and

{\ displaystyle \ Delta _ {1} ^ {2} -4 \ Delta _ {0} ^ {3} = - 27 \ Delta \,}

where is the above discriminant . Any of the complex third roots can be used for the third root occurring in.

{\ displaystyle \ Delta}

{\ displaystyle Q}

Special cases of the formula

If and must the sign of be chosen so that . ${\ displaystyle \ Delta \ neq 0}$ ${\ displaystyle \ Delta _ {0} = 0,}$ ${\ displaystyle {\ sqrt {\ Delta _ {1} ^ {2} -4 \ Delta _ {0} ^ {3}}} = {\ sqrt {\ Delta _ {1} ^ {2}}}}$ ${\ displaystyle Q \ neq 0,}$
If , the choice of the third root in the definition of must be changed so that this is always possible, except when the fourth degree polynomial can be factored as, which gives the solutions. ${\ displaystyle S = 0,}$ ${\ displaystyle Q}$ ${\ displaystyle S \ neq 0.}$ ${\ displaystyle \ left (x + {\ tfrac {b} {4a}} \ right) ^ {4}}$