6.1: Kuongeza na Ondoa Polynomials
Mwishoni mwa sehemu hii, utaweza:
- Kutambua polynomials, monomials, binomials, na trinomials
- Kuamua kiwango cha polynomials
- Ongeza na uondoe monomials
- Ongeza na uondoe polynomials
- Tathmini polynomial kwa thamani iliyotolewa
Kabla ya kuanza, fanya jaribio hili la utayari.
- Kurahisisha:8x+3x.
Kama amekosa tatizo hili, kupitia Zoezi 1.3.37. - Ondoa:(5n+8)−(2n−1).
Kama amekosa tatizo hili, kupitia Zoezi 1.10.52. - Andika katika fomu iliyopanuliwa:a5.
Ikiwa umekosa tatizo hili, kagua Zoezi 1.3.7.
Kutambua Polynomials, Monomials, Binomials na Trinomials
Umejifunza kwamba neno ni mara kwa mara au bidhaa ya vigezo vya mara kwa mara na moja au zaidi. Wakati ni wa fomuaxm, wapia mara kwa mara nam ni namba nzima, inaitwa monomial. Baadhi ya mifano ya monomial ni8,−2x2,4y3, na11z7.
Monomial ni neno la fomuaxm, ambapoa ni mara kwa mara nam ni nambari 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 monomials mbili au zaidi pamoja na kuongeza au kutoa, 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 b+14y2−7y+24x4+x3+8x2−9x+1 Monomial 148y2−9x3y5−13 Binomial a+74b−5y2−163x3−9x2 Trinomial x2−7x+129y2+2y−86m4−m3+8mz4+3z2−1
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.
Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine.
- 4y2−8y−6
- −5a4b2
- 2x5−5x3−3x+4
- 13−5m3
- q
- Jibu
-
Polynomial Number of terms Type (a) 4y2−8y−63 Trinomial (b) −5a4b21 Monomial (c) 2x5−5x3−9x2+3x+45 Ponomial (d) 13−5m32 Binomial (e) q1 Monomial
Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine:
- 5b
- 8y3−7y2−y−3
- −3x2−5x+9
- 81−4a2
- −5x6
- Jibu
-
- monomial
- polynomial
- ya trinomial
- binomial
- monomial
Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine:
- 27z3−8
- 12m3−5m2−2m
- 56
- 8x4−7x2−6x−5
- −n4
- Jibu
-
- binomial
- ya trinomial
- monomial
- polynomial
- monomial
Kuamua Shahada ya Polynomials
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, yaani, haina kutofautiana.
- 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.
Polynomial iko katika fomu ya kawaida wakati maneno ya polynomial yameandikwa katika utaratibu wa kushuka kwa digrii. Pata tabia ya kuandika neno kwa shahada ya juu kwanza.
Pata shahada ya polynomials zifuatazo.
- 10y
- 4x3−7x+5
- -15
- −8b2+9b−2
- 8xy2+2y
- Jibu
-
- 10yThe exponent of y is one. y=y1The degree is 1.
- 4x3−7x+5The highest degree of all the terms is 3.The degree is 3.
- −15The degree of a constant is 0.The degree is 0.
- −8b2+9b−2The highest degree of all the terms is 2.The degree is 2.
- 8xy2+2yThe highest degree of all the terms is 3.The degree is 3.
Pata shahada ya polynomials zifuatazo:
- -15b
- 10z4+4z2−5
- 12c5d4+9c3d9−7
- 3x2y−4x
- —9
- Jibu
-
- 1
- 4
- 12
- 3
- 0
Pata shahada ya polynomials zifuatazo:
- 52
- a4b−17a4
- 5x+6y+2z
- 3x2−5x+7
- −a3
- Jibu
-
- 0
- 5
- 1
- 2
- 3
Kuongeza na Ondoa Monomials
Umejifunza 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 mgawo.
Ongeza:25y2+15y2
- Jibu
-
25y2+15y2Combine like terms.40y2
Ongeza:12q2+9q2
- Jibu
-
21q2
Ongeza:−15c2+8c2
- Jibu
-
−7c2
Ondoa: 16p- (-7p)
- Jibu
-
16p−(−7p)Combine like terms.23p
Ondoa: 8m-m (-5m).
- Jibu
-
13m
Ondoa:−15z3−(−5z3)
- Jibu
-
−10z3
Kumbuka kwamba kama maneno lazima kuwa na vigezo sawa na exponents sawa.
Kurahisisha:c2+7d2−6c2
- Jibu
-
c2+7d2−6c2Combine like terms.−5c2+7d2
Ongeza:8y2+3z2−3y2
- Jibu
-
5y2+3z2
Ongeza:3m2+n2−7m2
- Jibu
-
−4m2+n2
Kurahisisha:u2v+5u2−3v2
- Jibu
-
\ (\ kuanza {safu} {ll} &u^ {2} v+5 u^ {2} -3 v^ {2}
\\ maandishi {2}\\ Nakala {Hakuna maneno kama ya kuchanganya.} & u^ {2} v+5 u^ {2} -3 v^ {2}\ mwisho {safu}\)
Kurahisisha:m2n2−8m2+4n2
- Jibu
-
Hakuna maneno kama hayo ya kuchanganya.
Kurahisisha:pq2−6p−5q2
- Jibu
-
Hakuna maneno kama hayo ya kuchanganya.
Kuongeza na Ondoa Polynomials
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:(5y2−3y+15)+(3y2−4y−11)
- Jibu
-
Tambua maneno kama hayo. Panga upya ili kupata maneno kama hayo pamoja. Kuchanganya kama maneno.
Pata jumla:(7x2−4x+5)+(x2−7x+3)
- Jibu
-
8x2−11x+1
Pata jumla:(14y2+6y−4)+(3y2+8y+5)
- Jibu
-
17y2+14y+1
Pata tofauti:(9w2−7w+5)−(2w2−4)
- Jibu
-
Kusambaza na kutambua maneno kama hayo. Panga upya masharti. Kuchanganya kama maneno.
Pata tofauti:(8x2+3x−19)−(7x2−14)
- Jibu
-
15x2+3x−5
Pata tofauti:(9b2−5b−4)−(3b2−5b−7)
- Jibu
-
6b2+3
Ondoa:(c2−4c+7) kutoka(7c2−5c+3)
- Jibu
-
Kusambaza na kutambua maneno kama hayo. Panga upya masharti. Kuchanganya kama maneno.
Ondoa:(5z2−6z−2) kutoka(7z2+6z−4)
- Jibu
-
2z2+12z−2
Ondoa:(x2−5x−8) kutoka(6x2+9x−1)
- Jibu
-
5x2+14x+7
Pata jumla:(u2−6uv+5v2)+(3u2+2uv)
- Jibu
-
(u2−6uv+5v2)+(3u2+2uv)Distribute.u2−6uv+5v2+3u2+2uvRearrange the terms, to put like terms togetheru2+3u2−6uv+2uv+5v2Combine like terms.4u2−4uv+5v2
Pata jumla:(3x2−4xy+5y2)+(2x2−xy)
- Jibu
-
5x2−5xy+5y2
Pata jumla:(2x2−3xy−2y2)+(5x2−3xy)
- Jibu
-
7x2−6xy−2y2
Pata tofauti:(p2+q2)−(p2+10pq−2q2)
- Jibu
-
(p2+q2)−(p2+10pq−2q2)Distribute.p2+q2−p2−10pq+2q2Rearrange the terms, to put like terms togetherp2−p2−10pq+q2+2q2Combine like terms.−10pq+3q2
Pata tofauti:(a2+b2)−(a2+5ab−6b2)
- Jibu
-
−5ab−5b2
Pata tofauti:(m2+n2)−(m2−7mn−3n2)
- Jibu
-
4n2+7mn
Kurahisisha:(a3−a2b)−(ab2+b3)+(a2b+ab2)
- Jibu
-
(a3−a2b)−(ab2+b3)+(a2b+ab2)Distribute.a3−a2b−ab2−b3+a2b+ab2Rearrange the terms, to put like terms togethera3−a2b+a2b−ab2+ab2−b3Combine like terms.a3−b3
Kurahisisha:(x3−x2y)−(xy2+y3)+(x2y+xy2)
- Jibu
-
x3−y3
Kurahisisha:(p3−p2q)+(pq2+q3)−(p2q+pq2)
- Jibu
-
p3−2p2q+q3
Tathmini Polynomial kwa Thamani iliyotolewa
Tayari tumejifunza jinsi ya kutathmini maneno. Kwa kuwa polynomials ni maneno, tutaweza kufuata taratibu hiyo ya kutathmini polynomial. Tutabadilisha thamani iliyotolewa kwa kutofautiana na kisha kurahisisha kutumia utaratibu wa shughuli.
Tathmini5x2−8x+4 wakati
- x=4
- x=—2
- x=0
- Jibu
-
1. x=4 Kurahisisha watetezi. Kuzidisha. Kurahisisha. 2. x=-1 Kurahisisha watetezi. Kuzidisha. Kurahisisha. 3. x=0 Kurahisisha watetezi. Kuzidisha. Kurahisisha.
Tathmini:3x2+2x−15 wakati
- x=3
- x=-5
- x=0
- Jibu
-
- 18
- 50
- -15
Tathmini:5z2−z−4 wakati
- z=—2
- z=0
- z=2
- Jibu
-
- 18
- —4
- 14
Polynomial−16t2+250 anatoa urefu wa mpira tt sekunde baada ya imeshuka kutoka 250 mguu mrefu jengo. Pata urefu baada ya sekunde t=2.
- Jibu
-
−16t2+250Substitute t = 2.−16(2)2+250Simplify −16⋅4+250Simplify −64+250Simplify 186After 2 seconds the height of the ball is 186 feet.
Polynomial−16t2+250 anatoa urefu wa mpira tt sekunde baada ya imeshuka kutoka 250 mguu mrefu jengo. Pata urefu baada ya sekunde t=0.
- Jibu
-
250
Polynomial−16t2+250 inatoa urefu wa mpira tt sekunde baada ya imeshuka kutoka 250 mguu mrefu jengo. Pata urefu baada ya sekunde t=3.
- Jibu
-
106
Polynomial6x2+15xy inatoa gharama, kwa dola, ya kuzalisha chombo mstatili ambao juu na chini ni mraba na upande x miguu na pande za urefu y miguu. Pata gharama ya kuzalisha sanduku yenye futi x=4 na y=6y=futi 6.
- Jibu
-
Kurahisisha. Kurahisisha. Kurahisisha. Gharama ya kuzalisha sanduku ni $456.
Polynomial6x2+15xy inatoa gharama, kwa dola, ya kuzalisha chombo mstatili ambao juu na chini ni mraba na upande x miguu na pande za urefu y miguu. Pata gharama ya kuzalisha sanduku yenye futi x=6 na y=futi 4.
- Jibu
-
$576
Polynomial6x2+15xy inatoa gharama, kwa dola, ya kuzalisha chombo mstatili ambao juu na chini ni mraba na upande x miguu na pande za urefu y miguu. Pata gharama ya kuzalisha sanduku yenye futi x=5 na y=futi 8.
- Jibu
-
$750
Dhana muhimu
- Monomials
- Monomial ni neno la fomuaxm, ambapo aa ni mara kwa mara na mm ni namba nzima
- Monomial ni neno la fomuaxm, ambapo aa ni mara kwa mara na mm ni namba nzima
- Polynomials
- polynomial -monomial, au monomials mbili au zaidi pamoja na kuongeza au kutoa 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.
- 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 kielelezo cha kutofautiana kwake.
- monomial
- Monomial ni neno la fomuaxm, ambapo a ni mara kwa mara na m ni namba nzima; monomial ina muda mmoja hasa.
- polynomial
- Polynomial ni monomial, au monomials mbili au zaidi pamoja na kuongeza au kuondoa.
- fomu ya kawaida
- 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.