Faktorisasi: Perbedaan antara revisi
Konten dihapus Konten ditambahkan
JohnThorne (bicara | kontrib) Tidak ada ringkasan suntingan |
k ejaan, replaced: obyek → objek (3) |
||
Baris 1:
[[File:Factorisatie.svg|thumb|rught|Polinomial ''x''<sup>2</sup> + ''cx'' + ''d'', di mana ''a + b = c'' dan ''ab = d'', dapat difaktorisasi menjadi (''x + a'')(''x + b'').]]
'''Faktorisasi''' dalam [[matematika]] adalah dekomposisi suatu
Tujuan faktorisasi biasanya untuk mereduksi sesuatu menjadi "blok pembangun dasar" (“basic building blocks”), seperti bilangan-bilangan prima, atau polinomial menjadi [[:en:irreducible polynomial|polinomial tak tereduksi]]. Faktorisasi integers diatur oleh [[:en:fundamental theorem of arithmetic|rumus dasar aritmetika]] dan [[:en:polynomial factorization|faktorisasi polinomial]] diatur oleh [[:en:fundamental theorem of algebra|teorema dasar aljabar]]. [[:en:Viète's formulas|Rumus-rumus Viète]] mengkaitkan [[koefisien]]-koefisien suatu polinomial pada akar-akarnya.
Baris 36:
A method that is sometimes useful, but not guaranteed to work, is factoring by grouping.
Factoring by grouping is done by placing the terms in the polynomial into two or more groups, where each group can be factored by a known method. The results of these partial factorizations can sometimes be combined to give a factorization of the original expression.
For example, to factor the polynomial
Baris 44:
# again factor out the binomial common factor, <math>(x+5)(4x+3y).\,</math>
While grouping may not lead to a factorization in general, if the polynomial expression to be factored consists of four terms and is the result of multiplying two binomial expressions (by the [[FOIL method]] for instance), then the grouping technique can lead to a factorization, as in the above example.
====Using the factor theorem====
{{main|Factor theorem}}
For a univariate polynomial, ''p''(''x''), the [[factor theorem]] states that ''a'' is a ''root'' of the polynomial (that is, ''p''(''a'') = 0, also called a ''zero'' of the polynomial) if and only if (''x'' - ''a'') is a factor of ''p''(''x''). The other factor in such a factorization of ''p''(''x'') can be obtained by [[polynomial long division]] or [[synthetic division]].
For example, consider the polynomial <math>x^3 - 3x + 2.</math> By inspection we see that 1 is a root of this polynomial (observe that the coefficients add up to 0), so (''x'' - 1) is a factor of the polynomial. By long division we have <math>x^3 - 3x + 2 = (x - 1)(x^2 + x - 2).</math>
Baris 86:
So, we have:
:<math>a^2x^2 + abx +ac = (ax+r)(ax+s),</math>
where ''rs'' = ''ac'' and ''r'' + ''s'' = ''b''. The ''ac method'' for factoring the quadratic polynomial is to find ''r'' and ''s'', the two factors of the number ''ac'' whose sum is ''b'' and then use them in [[#ac|the factorization formula of the original quadratic above]].
As an example consider the quadratic polynomial:
Baris 189:
where <math>\alpha</math> and <math>\beta</math> are the two [[root of a function|roots]] of the polynomial, found with the [[quadratic formula]].
The quadratic formula is valid for all polynomials with coefficients in any [[Field (mathematics)|field]] (in particular, the real or complex numbers) except those that have [[Characteristic (algebra)|characteristic]] two.<ref>In these fields 2 = 0 so the division in the formula is not valid. There are other ways to find roots of quadratic equations over these fields.</ref>
There are also formulas for cubic and quartic polynomials which can be used in the same way. However, there are no formulas in terms of the coefficients that exist for higher degree (univariate) polynomials by the [[Abel-Ruffini theorem]].
Baris 259:
* [[Wolfram Alpha]] [http://www.wolframalpha.com/input/?i=Factor%20-2006+%2B+1155+x+-+78+x^2+%2B+x^3 can factorize too].
[[
[[
| |||