5.2: Kuongeza na Ondoa Polynomials
- Page ID
- 176039
Mwishoni mwa sehemu hii, utaweza:
- Kuamua kiwango cha polynomials
- Ongeza na uondoe polynomials
- Tathmini kazi ya polynomial kwa thamani iliyotolewa
- Ongeza na uondoe kazi za polynomial
Kuamua Shahada ya Polynomials
Tumejifunza kwamba neno ni mara kwa mara au bidhaa ya vigezo mara kwa mara na moja au zaidi. Monomial ni kujieleza algebraic na neno moja. Wakati ni wa fomu\(ax^m\), wapi\(a\) mara kwa mara na\(m\) ni namba nzima, inaitwa monomial katika variable moja. Baadhi ya mifano ya monomial katika variable moja ni. Monomials pia inaweza kuwa na variable zaidi ya moja kama vile na\(−4a^2b^3c^2.\)
Monomial ni kujieleza algebraic na neno moja. Monomial katika variable moja ni neno la fomu\(ax^m\), ambapo\(a\) ni mara kwa mara na\(m\) ni namba nzima.
Monomial, au monomials mbili au zaidi pamoja na kuongeza au kuondoa, ni polynomial. Baadhi ya polynomials wana majina maalum, kulingana na idadi ya maneno. Monomial ni polynomial yenye muda mmoja. Binomial ina maneno mawili, na trinomial ina maneno matatu hasa. Hakuna majina maalum ya polynomials yenye maneno zaidi ya tatu.
- polynomial -monomial, au maneno mawili au zaidi ya algebraic pamoja na kuongeza au kuondoa ni polynomial.
- monomial -polynomial na neno moja hasa inaitwa monomial.
- binomial —polynomial yenye maneno mawili hasa inaitwa binomial.
- trinomial -polynomial na maneno matatu hasa inaitwa trinomial.
Hapa ni baadhi ya mifano ya polynomials.
| Polynomial | \(y+1\) | \(4a^2−7ab+2b^2\) | \(4x^4+x^3+8x^2−9x+1\) | |
|---|---|---|---|---|
| Monomial | \(14\) | \(8y^2\) | \(−9x^3y^5\) | \(−13a^3b^2c\) |
| Binomial | \(a+7ba+7b\) | \(4x^2−y^2\) | \(y^2−16\) | \(3p^3q−9p^2q\) |
| Trinomial | \(x^2−7x+12\) | \(9m^2+2mn−8n^2\) | \(6k^4−k^3+8k\) | \(z^4+3z^2−1\) |
Angalia kwamba kila monomial, binomial, na trinomial pia ni polynomial. Wao ni wanachama maalum wa “familia” ya polynomials na hivyo wana majina maalum. Tunatumia maneno monomial, binomial, na trinomial wakati akimaanisha polynomials hizi maalum na tu wito wote polynomials wengine.
Kiwango cha polynomial na kiwango cha masharti yake ni kuamua na exponents ya variable. Monomial ambayo haina kutofautiana, tu mara kwa mara, ni kesi maalum. Kiwango cha mara kwa mara ni 0.
- Kiwango cha neno ni jumla ya watazamaji wa vigezo vyake.
- Kiwango cha mara kwa mara ni 0.
- Kiwango cha polynomial ni kiwango cha juu cha masharti yake yote.
Hebu tuone jinsi hii inavyofanya kazi kwa kuangalia polynomials kadhaa. Tutachukua hatua kwa hatua, kuanzia na monomials, na kisha kuendelea kwa polynomials na maneno zaidi. Hebu tuanze kwa kuangalia monomial. Monomial\(8ab^2\) ina vigezo viwili\(a\) na\(b\). Ili kupata shahada tunahitaji kupata jumla ya exponents. a variable hana exponent imeandikwa, lakini kumbuka kwamba ina maana exponent ni 1. exponent ya\(b\) ni 2. Jumla ya exponents, 1+2,1+2, ni 3 hivyo shahada ni 3.
Hapa kuna mifano ya ziada.
Kufanya kazi na polynomials ni rahisi wakati unapoorodhesha maneno katika utaratibu wa kushuka kwa digrii. Wakati polynomial imeandikwa kwa njia hii, inasemekana kuwa katika fomu ya kawaida ya polynomial. Pata tabia ya kuandika neno kwa shahada ya juu kwanza.
Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine. Kisha, pata shahada ya kila polynomial.
- \(7y2−5y+3\)
- \(−2a^4b^2\)
- \(3x5−4x3−6x2+x−8\)
- \(2y−8xy^3\)
- \(15\)
- Jibu
-
Polynomial Idadi ya maneno Aina Shahada ya maneno Shahada ya polynomial ⓐ \(7y^2−5y+3\) 3 Trinomial 2, 1, 0 2 ⓑ \(−2a^4b^2−2a^4b^2\) 1 Monomial 4, 2 6 ⓒ \(3x5−4x3−6x2+x−8\) 5 Polynomial 5, 3, 2, 1, 0 5 ⓓ \(2y−8xy^3\) 2 Binomial 1, 4 4 ⓔ \(15\) 1 Monomial 0 0
Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine. Kisha, pata shahada ya kila polynomial.
- \(−5\)
- \(8y^3−7y^2−y−3\)
- \(−3x^2y−5xy+9xy^3\)
- \(81m^2−4n^2\)
- \(−3x^6y^3z\)
- Jibu
-
monomial, 0
- Jibu b
-
polynomial, 3
- Jibu c
-
trinomial, 3
- Jibu d
-
binomial, 2
- Jibu b
-
monomial, 10
Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine. Kisha, pata shahada ya kila polynomial.
- \(64k^3−8\)
- \(9m^3+4m^2−2\)
- \(56\)
- \(8a^4−7a^3b−6a^2b^2−4ab^3+7b^4\)
- \(-p^4q^3\)
- Jibu
-
ⓐ binomial, 3 ⓑ trinomial, 3 ⓒ monomial, 0 ⓓ polynomial, 4 ⓔ monomial, 7
Kuongeza na Ondoa Polynomials
Tumejifunza jinsi ya kurahisisha maneno kwa kuchanganya maneno kama hayo. Kumbuka, kama maneno lazima kuwa na vigezo sawa na exponent sawa. Kwa kuwa monomials ni maneno, kuongeza na kuondoa monomials ni sawa na kuchanganya maneno kama hayo. Ikiwa monomials ni kama maneno, tunawachanganya tu kwa kuongeza au kuondoa coefficients.
Ongeza au uondoe:
- \(25y^2+15y^2\)
- \(16pq^3−(−7pq^3)\).
- Jibu
-
\( \begin{array} {ll} {} &{25y^2+15y^2} \\ {\text{Combine like terms.}} &{40y^2} \\ \end{array} \nonumber \)
- Jibu b
-
\( \begin{array} {ll} {} &{16pq^3−(−7pq^3)} \\ {\text{Combine like terms.}} &{23pq^3} \\ \end{array} \nonumber \)
Ongeza au uondoe:
- \(12q^2+9q^2\)
- \(8mn^3−(−5mn^3)\).
- Jibu
-
ⓐ\(21q^2\) ⓑ\(13mn^3\)
Ongeza au uondoe:
- \(−15c^2+8c^2\)
- \(−15y^2z^3−(−5y^2z^3)\)
- Jibu
-
ⓐ\(−7c^2\) ⓑ\(−10y^2z^3\)
Kumbuka kwamba kama maneno lazima kuwa na vigezo sawa na exponents sawa.
Kurahisisha:
- \(a^2+7b^2−6a^2\)
- \(u^2v+5u^2−3v^2\)
- Jibu
-
ⓐ Kuchanganya kama maneno.
\(a^2+7b^2−6a^2 \;=\; −5a^2+7b^2\)ⓑ Hakuna maneno kama hayo ya kuchanganya. Katika kesi hiyo, polynomial haibadilika.
\(u^2v+5u^2−3v^2\)
Ongeza:
- \(8y^2+3z^2−3y^2\)
- \(m^2n^2−8m^2+4n^2\)
- Jibu
-
ⓐ\(5y^2+3z^2\)
ⓑ\(m^2n^2−8m^2+4n^2\)
Ongeza:
- \(3m^2+n^2−7m^2\)
- \(pq^2−6p−5q^2\)
- Jibu
-
ⓐ\(−4m^2+n^2\)
ⓑ\(pq^2−6p−5q^2\)
Tunaweza kufikiria kuongeza na kutoa polynomials kama tu kuongeza na kutoa mfululizo wa monomials. Angalia kwa masharti kama-wale walio na vigezo sawa na exponent sawa. Mali ya Comutative inatuwezesha kupanga upya masharti ili kuweka maneno kama pamoja.
Pata jumla:\((7y^2−2y+9)\;+\;(4y^2−8y−7)\).
- Jibu
-
\ (\ kuanza {align*} &\ Nakala {Tambua kama maneno.} & (\ kusisitiza {\ kusisitiza {7y ^ 2}}} -\ kusisitiza {2y} +9) + (\ kusisitiza {\ kusisitiza {4y ^ 2}}} -\ kusisitiza {8y} -7)\\ [6pt]
&\ maandishi {Andika upya bila mabano,}\\
&\ maandishi {upya ili kupata maneno kama hayo pamoja.} & &\ kusisitiza {\ kusisitiza {7y ^ 2+4y ^ 2}}} -\ kusisitiza {2y-8y} +9:7\\ [6pt]
&\ maandishi {Jumuisha kama maneno.} & & 11y ^ 2,110y+2\ mwisho {align*}\)
Pata jumla:\( (7x^2−4x+5)\;+\;(x^2−7x+3)\)
- Jibu
-
\(8x^2−11x+8\)
Pata jumla:\((14y^2+6y−4)\;+\;(3y^2+8y+5)\)
- Jibu
-
\(17y^2+14y+1\)
Kuwa makini na ishara kama wewe kusambaza wakati kuondoa polynomials katika mfano unaofuata.
Pata tofauti:\((9w^2−7w+5)\;−\;(2w^2−4)\)
- Jibu
-
\ (\ kuanza {align*} & & & (9w ^ 2,17w+5)\; -\; (2w ^ 2—4)\\ [6pt]
&\ maandishi {Kusambaza na kutambua maneno kama.} & &\ kusisitiza {\ kusisitiza {9w ^ 2}}} -\ kusisitiza {7w} +5-\ kusisitiza {\ kusisitiza {2w ^ 2}} +4\\ [6pt]
&\ maandishi {Panga upya masharti.} & &\ kusisitiza {\ kusisitiza {9w ^ 2-2w ^ 2}}} -\ kusisitiza {7w} +5+4\\ [6pt]
&\ maandishi {Changanya kama maneno.} & & 7w ^ 2,17w+9\ mwisho {align*}\)
Pata tofauti:\((8x^2+3x−19)\;−\;(7x^2−14)\)
- Jibu
-
\(x^2+3x−5\)
Pata tofauti:\((9b^2−5b−4)\;−\;(3b^2−5b−7)\)
- Jibu
-
\(6b^2+3\)
Ondoa\((p^2+10pq−2q^2)\) kutoka\((p^2+q^2)\).
- Jibu
-
\ (\ kuanza {align*} & & & (p ^ 2+q ^ 2)\; -\; (p^2+10pq-2q ^ 2)\\ [6pt]
&\ maandishi {Kusambaza na kutambua maneno kama.} & &\ kusisitiza {\ kusisitiza {p^2}} +\ kusisitiza {q ^ 2} -\ kusisitiza {\ kusisitiza {p ^ 2}} -10pq +\ kusisitiza {2q ^ 2}\ [6pt]
&\ maandishi {Panga upya maneno, kuweka maneno kama pamoja.} & &\ kusisitiza {\ kusisitiza {p ^ 2-p^2}} -10pq +\ kusisitiza {q ^ 2 + 2q ^ 2}\\ [6pt]
&\ maandishi {Jumuisha kama maneno.} & -10pq+3q ^ 2\ mwisho {align*}\)
Ondoa\((a^2+5ab−6b^2)\) kutoka\((a^2+b^2)\)
- Jibu
-
\(−5ab+7b^2\)
Ondoa\((m^2−7mn−3n^2)\) kutoka\((m^2+n^2)\).
- Jibu
-
7mn+4n ^ 2
Pata jumla:\((u^2−6uv+5v^2)\;+\;(3u^2+2uv)\)
- Jibu
-
\ (\ kuanza {align*} & & & (u ^ 2,16uv+5v ^ 2)\; +\; (3u ^ 2+2uv)\\ [6pt]
&\ maandishi {Kusambaza na kutambua kama maneno.} & &\ kusisitiza {\ kusisitiza {u ^ 2}} -\ kusisitiza {6uv} +5v ^ 2+\ kusisitiza {\ kusisitiza {3u ^ 2}} +\ kusisitiza {2uv}\ [6pt]
&\ maandishi {Panga upya maneno ili kuweka maneno kama pamoja.} & &\ kusisitiza {\ kusisitiza {u ^ 2}} +\ kusisitiza {\ kusisitiza {3u ^ 2}} -\ kusisitiza {6uv} +\ kusisitiza {2uv} +5v ^ 2\\ [6pt]
&\ maandishi {Changanya kama maneno.} & & 4u ^ 2,14uv+5v ^ 2\ mwisho {align*}\)
Pata jumla:\((3x^2−4xy+5y^2)\;+\;(2x^2−xy)\)
- Jibu
-
\(5x^2−5xy+5y^2\)
Pata jumla:\((2x^2−3xy−2y^2)\;+\;(5x^2−3xy)\)
- Jibu
-
\(7x^2−6xy−2y^2\)
Tunapoongeza na kuondoa polynomials zaidi ya mbili, mchakato huo ni sawa.
Kurahisisha:\((a^3−a^2b)\;−\;(ab^2+b^3)\;+\;(a^2b+ab^2)\)
- Jibu
-
\ (\ kuanza {align*} & & & (a ^ 3,1a ^ 2b)\; -\; (ab^2+b ^ 3)\; +\; (a ^ 2b+ab^2)\\ [6pt]
&\ maandishi {Kusambaza} & a^3—a ^ 2b - b ^ 3 + a^ 2b+ab^2\ [6pt]
&\ maandishi {Panga upya masharti ili kuweka kama maneno pamoja.} & a ^ 3,1a^2b + a^2b - ab^2 + ab^2 - b ^ 3\\ [6pt]
&\ maandishi {Jumuisha kama maneno.} & a ^ 3,1b ^ 3\ mwisho {align*}\)
Kurahisisha:\((x^3−x^2y)\;−\;(xy^2+y^3)\;+\;(x^2y+xy^2)\)
- Jibu
-
\(x^3+y^3\)
Kurahisisha:\((p^3−p^2q)\;+\;(pq^2+q^3)\;−\;(p^2q+pq^2)\)
- Jibu
-
\(p^3−3p^2q+q^3\)
Tathmini Kazi ya Polynomial kwa Thamani iliyotolewa
Kazi ya polynomial ni kazi inayofafanuliwa na polynomial. Kwa mfano,\(f(x)=x^2+5x+6\) na\(g(x)=3x−4\) ni kazi polynomial, kwa sababu\(x^2+5x+6\) na\(3x−4\) ni polynomials.
Kazi ya polynomial ni kazi ambayo maadili mbalimbali hufafanuliwa na polynomial.
Katika Grafu na Kazi, ambapo sisi kwanza ilianzisha kazi, tulijifunza kwamba kutathmini kazi ina maana ya kupata thamani ya\(f(x)\) kwa thamani fulani ya\(x\). Ili kutathmini kazi ya polynomial, tutabadilisha thamani iliyotolewa kwa kutofautiana na kisha kurahisisha kutumia utaratibu wa shughuli.
Kwa kazi\(f(x)=5x^2−8x+4\) kupata:
- \(f(4)\)
- \(f(−2)\)
- \(f(0)\).
- Jibu
-
ⓐ
Kurahisisha watetezi. 
Kuzidisha. 
Kurahisisha. 
ⓑ
Kurahisisha watetezi. 
Kuzidisha. 
Kurahisisha. 
ⓒ
Kurahisisha watetezi. 
Kuzidisha. 
Kwa kazi\(f(x)=3x^2+2x−15\), tafuta
- \(f(3)\)
- \(f(−5)\)
- \(f(0)\).
- Jibu
-
ⓐ 18 ⓑ 50 ⓒ\(−15\)
Kwa kazi\(g(x)=5x^2−x−4\), tafuta
- \(g(−2)\)
- \(g(−1)\)
- \(g(0)\).
- Jibu
-
ⓐ 20 ⓑ 2 ⓒ\(−4\)
Kazi za polynomial zinazofanana na moja katika mfano unaofuata zinatumiwa katika nyanja nyingi ili kuamua urefu wa kitu kwa wakati fulani baada ya kupangwa ndani ya hewa. Polynomial katika kazi inayofuata hutumiwa mahsusi kwa kuacha kitu kutoka 250 ft.
Kazi ya polynomial\(h(t)=−16t^2+250\) inatoa urefu wa mpira t sekunde baada ya kushuka kutoka jengo la urefu wa 250 futi. Pata urefu baada ya\(t=2\) sekunde.
- Jibu
-
\( \begin{array} {ll} {} &{h(t)=−16t^2+250} \\ {} &{} \\ {\text{To find }h(2)\text{, substitute }t=2.} &{h(2)=−16(2)^2+250} \\ {\text{Simplify.}} &{h(2)=−16·4+250} \\ {} &{}\\ {\text{Simplify.}} &{h(2)=−64+250} \\ {} &{} \\ {\text{Simplify.}} &{h(2)=186} \\ {} &{\text{After 2 seconds the height of the ball is 186 feet.}} \\ \end{array} \nonumber \)
Kazi ya polynomial\(h(t)=−16t^2+150\) inatoa urefu wa jiwe t sekunde baada ya kushuka kutoka kwenye mwamba mrefu wa futi 150. Pata urefu baada ya\(t=0\) sekunde (urefu wa awali wa kitu).
- Jibu
-
Urefu ni\(150\) miguu.
Kazi ya polynomial\(h(t)=−16t^2+175\) inatoa urefu wa mpira t sekunde baada ya kushuka kutoka daraja la urefu wa 175-futi. Pata urefu baada ya\(t=3\) sekunde.
- Jibu
-
Urefu ni\(31\) miguu.
Ongeza na Ondoa Kazi za Polynomial
Kama vile polynomials inaweza kuongezwa na kuondolewa, kazi za polynomial pia zinaweza kuongezwa na kuondolewa.
Kwa ajili ya kazi\(f(x)\) na\(g(x)\),
\[(f+g)(x)=f(x)+g(x)\]
\[(f−g)(x)=f(x)−g(x)\]
Kwa kazi\(f(x)=3x^2−5x+7\) na\(g(x)=x^2−4x−3\), tafuta:
- \((f+g)(x)\)
- \((f+g)(3)\)
- \((f−g)(x)\)
- \((f−g)(−2)\).
- Jibu
-
ⓐ
Andika upya bila mabano. 
Weka maneno kama pamoja. 
Kuchanganya kama maneno. 
ⓑ Katika sehemu (a) tulipata\((f+g)(x)\) na sasa tunaulizwa kupata\((f+g)(3)\).
\( \begin{array} {ll} {} &{(f+g)(x)=4x^2−9x+4} \\ {} &{} \\ {\text{To find }(f+g)\space(3),\text{ substitute }x=3.} &{(f+g)(3)=4(3)^2−9·3+4} \\ {} &{} \\ {} &{(f+g)(3)=4·9−9·3+4} \\ {} &{} \\ {} &{(f+g)(3)=36−27+4} \\ \end{array} \nonumber \)
Taarifa kwamba tunaweza kuwa kupatikana\((f+g)(3)\) kwa kwanza kutafuta maadili ya\(f(3)\) na\(g(3)\) tofauti na kisha kuongeza matokeo.
Kupata\(f(3)\). 
Kupata\(g(3)\). 
Kupata\((f+g)(3)\). 
ⓒ
Andika upya bila mabano. 
Weka maneno kama pamoja. 
Kuchanganya kama maneno. 
ⓓ
Kwa ajili ya kazi\(f(x)=2x^2−4x+3\) na\(g(x)=x^2−2x−6\), kupata: ⓐ\((f+g)(x)\) ⓑ\((f+g)(3)\) ⓒ\((f−g)(x)\) ⓓ\((f−g)(−2)\).
- Jibu
-
ⓐ\((f+g)(x)=3x^2−6x−3\)
ⓑ\((f+g)(3)=6\)
ⓒ\((f−g)(x)=x^2−2x+9\)
ⓓ\((f−g)(−2)=17\)
Kwa ajili ya kazi\(f(x)=5x^2−4x−1\) na\(g(x)=x^2+3x+8\), kupata ⓐ\((f+g)(x)\) ⓑ\((f+g)(3)\) ⓒ\((f−g)(x)\) ⓓ\((f−g)(−2)\).
- Jibu
-
ⓐ\((f+g)(x)=6x^2−x+7\)
ⓑ\((f+g)(3)=58\)
ⓒ\((f−g)(x)=4x^2−7x−9\)
ⓓ\((f−g)(−2)=21\)
Fikia rasilimali hii ya mtandaoni kwa maelekezo ya ziada na mazoezi na kuongeza na kutoa polynomials.
- Kuongeza na Kutoa Polynomials
Dhana muhimu
- Monomial
- Monomial ni kujieleza algebraic na neno moja.
- Monomial katika variable moja ni neno la fomu axm, axm, ambapo a ni mara kwa mara na m ni namba nzima.
- Polynomials
- Polynomial -monomial, au maneno mawili au zaidi ya algebraic pamoja na kuongeza au kuondoa ni polynomial.
- monomial -polynomial na neno moja hasa inaitwa monomial.
- binomial — Polynomial na maneno mawili hasa inaitwa binomial.
- trinomial -polynomial na maneno matatu hasa inaitwa trinomial.
- Shahada ya Polynomial
- Kiwango cha neno ni jumla ya watazamaji wa vigezo vyake.
- Kiwango cha mara kwa mara ni 0.
- Kiwango cha polynomial ni kiwango cha juu cha masharti yake yote.
faharasa
- binomial
- Binomial ni polynomial na maneno mawili hasa.
- shahada ya mara kwa mara
- Kiwango cha mara kwa mara yoyote ni 0.
- shahada ya polynomial
- Kiwango cha polynomial ni kiwango cha juu cha masharti yake yote.
- shahada ya muda
- Kiwango cha neno ni jumla ya watazamaji wa vigezo vyake.
- monomial
- Monomial ni kujieleza algebraic na neno moja. Monomial katika variable moja ni neno la fomu axm, axm, ambapo a ni mara kwa mara na m ni namba nzima.
- polynomial
- Monomial au mbili au zaidi monomials pamoja na kuongeza au kuondoa ni polynomial.
- aina ya kawaida ya polynomial
- Polynomial iko katika fomu ya kawaida wakati maneno ya polynomial yameandikwa katika utaratibu wa kushuka kwa digrii.
- ya trinomial
- Trinomial ni polynomial na maneno matatu hasa.
- kazi ya polynomial
- Kazi ya polynomial ni kazi ambayo maadili mbalimbali hufafanuliwa na polynomial.