Qhov txawv ntawm Polynomial thiab Monomial

2024 Tus sau: Alex Aldridge | [email protected]. Kawg hloov kho: 2023-12-17 13:40

Polynomial vs Monomial

A polynomial yog txhais raws li kev qhia lej muab los ntawm cov ntsiab lus tsim los ntawm cov khoom sib txawv thiab cov coefficients. Yog hais tias qhov kev qhia muaj xws li ib tug kuj sib txawv, lub polynomial hu ua univariate, thiab yog hais tias qhov kev qhia muaj ob los yog ntau tshaj variables, nws yog multivariate.

Ib tug univariate polynomial feem ntau symbolized li P(x) yog muab los ntawm;

P(x)=a_nxⁿ + a_n-1 x ^n-1 + a_n-2x^n-2 +⋯+ a₀; qhov twg, x, a₀,₁, a₂,₃, a₄, … a_n ∈ R and n ∈ Z₀⁺

[Rau qhov kev qhia los ua ib lub npe, nws qhov sib txawv yuav tsum yog qhov sib txawv tiag tiag thiab cov coefficient kuj yog tiag. Thiab cov exponents yuav tsum tsis yog-negative integer

Polynomials feem ntau txawv los ntawm lub zog siab tshaj plaws ntawm cov ntsiab lus hauv polynomial thaum nws nyob hauv daim ntawv canonical, uas yog hu ua degree (lossis xaj) ntawm polynomial. Yog tias lub zog siab tshaj plaws ntawm ib lub sijhawm yog n, nws yog lub npe hu ua n^th degree polynomial [piv txwv li, Yog n=2, nws yog qhov kev txiav txim thib ob polynomial; yog n=3, nws yog 3^{rd xaj polynomial].}

Polynomial functions yog cov haujlwm uas qhov kev sib raug zoo ntawm tus sau-co-domain tau muab los ntawm polynomial. Ib qho kev ua haujlwm plaub yog qhov kev txiav txim thib ob polynomial muaj nuj nqi. Polynomial equation yog ib qho kev sib npaug uas ob los yog ntau tshaj polynomials sib npaug [yog tias qhov sib npaug zoo li P=Q, P thiab Q yog polynomials]. Lawv kuj hu ua algebraic equations.

Ib lub sij hawm ntawm lub polynomial yog monomial. Hauv lwm lo lus, lub ntsiab lus ntawm polynomial tuaj yeem suav tias yog monomial. Nws muaj daim ntawv a_nx. Ib qho kev qhia nrog ob lub monomial yog lub npe hu ua binomial, thiab nrog peb lo lus hu ua trinomial [binomials ⇒ a_n xⁿ + b _n y, trinomial ⇒ a_nxⁿ + b_n yⁿ + c_nz^].

Polynomial yog qhov tshwj xeeb ntawm kev qhia lej thiab muaj ntau yam khoom tseem ceeb. Cov ntsiab lus ntawm polynomials yog ib tug polynomial. Cov khoom ntawm polynomials yog polynomial. Kev sib sau ntawm ib tug polynomial yog ib tug polynomial. Qhov sib txawv ntawm polynomials tsim polynomials.

Tsis tas li, cov npe ntau tuaj yeem siv los kwv yees lwm yam haujlwm uas siv txoj hauv kev tshwj xeeb xws li Taylor's series. Piv txwv li sin x, cos x, e^{x tuaj yeem kwv yees siv ntau lub zog. Hauv kev txheeb cais, kev sib raug zoo ntawm qhov sib txawv yog kwv yees siv polynomials los ntawm kev nrhiav qhov zoo tshaj plaws haum polynomial thiab txiav txim siab cov coefficients tsim nyog.}

Tus quotient ntawm ob polynomials ua rau muaj nuj nqi (x)=[P(x)] / [Q(x)], qhov twg Q(x)≠0.

Sib pauv cov coefficient xws li a₀ ⇌ a_n, a₁ ⇌ a _n-1, a₂ ⇌ a_{n-2, thiab lwm yam, qhov sib npaug ntawm ntau yam, uas nws cov hauv paus hniav yog qhov sib txuam ntawm tus thawj, tau.}

Qhov txawv ntawm Polynomial thiab Monomial yog dab tsi?

• Ib qho kev qhia lej tsim los ntawm cov khoom lag luam ntawm cov coefficients thiab qhov sib txawv thiab kev nthuav tawm ntawm qhov sib txawv yog hu ua monomial. Cov exponents tsis yog qhov tsis zoo, thiab qhov sib txawv thiab cov coefficients yog tiag.

• Ib tug polynomial yog ib qho kev qhia lej tsim los ntawm cov lej ntawm monomial. Yog li ntawd, peb tuaj yeem hais tias monomial yog summands ntawm polynomials lossis ib lub sijhawm ntawm polynomial yog monomial.

• Monomials tsis tuaj yeem muaj qhov sib ntxiv lossis rho tawm ntawm qhov sib txawv.

• Degree of the polynomials is the degree of the siab tshaj monomial.