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6.1: Kuongeza na Ondoa Polynomials

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    Malengo ya kujifunza

    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
    Quiz

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. Kurahisisha:\(8x+3x\).
      Kama amekosa tatizo hili, kupitia Zoezi 1.3.37.
    2. Ondoa:\((5n+8)−(2n−1)\).
      Kama amekosa tatizo hili, kupitia Zoezi 1.10.52.
    3. Andika katika fomu iliyopanuliwa:\(a^{5}\).
      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 fomu\(ax^{m}\), wapi\(a\) mara kwa mara na\(m\) ni namba nzima, inaitwa monomial. Baadhi ya mifano ya monomial ni\(8,−2x^{2},4y^{3}\), na\(11z^{7}\).

    Ufafanuzi: Monomials

    Monomial ni neno la fomu\(ax^{m}\), ambapo\(a\) ni mara kwa mara na\(m\) 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.

    ufafanuzi: 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.

    Hapa ni baadhi ya mifano ya polynomials.

    \[\begin{array}{lllll}{\text { Polynomial }} & {b+1} &{4 y^{2}-7 y+2} & {4 x^{4}+x^{3}+8 x^{2}-9 x+1} \\ {\text { Monomial }} & {14} & {8 y^{2}} & {-9 x^{3} y^{5}} & {-13}\\ {\text { Binomial }} & {a+7}&{4 b-5} & {y^{2}-16}& {3 x^{3}-9 x^{2}} \\ {\text { Trinomial }} & {x^{2}-7 x+12} & {9 y^{2}+2 y-8} & {6 m^{4}-m^{3}+8 m}&{z^{4}+3 z^{2}-1} \end{array} \nonumber\]

    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.

    Mfano\(\PageIndex{1}\)

    Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine.

    1. \(4y^{2}−8y−6\)
    2. \(−5a^{4}b^{2}\)
    3. \(2x^{5}−5x^{3}−3x + 4\)
    4. \(13−5m^{3}\)
    5. q
    Jibu

    \(\begin{array}{lll}&{\text { Polynomial }} & {\text { Number of terms }} & {\text { Type }} \\ {\text { (a) }} & {4 y^{2}-8 y-6} & {3} & {\text { Trinomial }} \\ {\text { (b) }} & {-5 a^{4} b^{2}} & {1} & {\text { Monomial }} \\ {\text { (c) }} & {2 x^{5}-5 x^{3}-9 x^{2}+3 x+4} & {5} & {\text { Ponomial }} \\ {\text { (d) }} & {13-5 m^{3}} & {2} & {\text { Binomial }} \\ {\text { (e) }} & {q} & {1} & {\text { Monomial }}\end{array}\)

    Mfano\(\PageIndex{2}\)

    Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine:

    1. 5b
    2. \(8 y^{3}-7 y^{2}-y-3\)
    3. \(-3 x^{2}-5 x+9\)
    4. \(81-4 a^{2}\)
    5. \(-5 x^{6}\)
    Jibu
    1. monomial
    2. polynomial
    3. ya trinomial
    4. binomial
    5. monomial
    Mfano\(\PageIndex{3}\)

    Kuamua kama kila polynomial ni monomial, binomial, trinomial, au polynomial nyingine:

    1. \(27 z^{3}-8\)
    2. \(12 m^{3}-5 m^{2}-2 m\)
    3. \(\frac{5}{6}\)
    4. \(8 x^{4}-7 x^{2}-6 x-5\)
    5. \(-n^{4}\)
    Jibu
    1. binomial
    2. ya trinomial
    3. monomial
    4. polynomial
    5. 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.

    Ufafanuzi: 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.

    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.


    Jedwali hili lina safu 11 na nguzo 5. Safu ya kwanza ni safu ya kichwa, na inaita kila mstari. Mstari wa kwanza huitwa “Monomial,” na kila kiini katika mstari huu kina monomial tofauti. Mstari wa pili huitwa “Shahada,” na kila kiini katika mstari huu kina kiwango cha monomial juu yake. Kiwango cha 14 ni 0, kiwango cha mraba 8y ni 2, kiwango cha 9x hasi cubed y kwa nguvu ya tano ni 8, na kiwango cha 13a hasi ni 1. Mstari wa tatu huitwa “Binomial,” na kila kiini katika mstari huu kina binomial tofauti. Mstari wa nne huitwa “Shahada ya kila neno,” na kila kiini kina digrii za maneno mawili katika binomial juu yake. Mstari wa tano huitwa “Shahada ya polynomial,” na kila kiini kina kiwango cha binomial kwa ujumla.” Daraja la maneno katika plus 7 ni 0 na 1, na kiwango cha binomial nzima ni 1. Daraja la maneno katika 4b squared minus 5b ni 2 na 1, na kiwango cha binomial nzima ni 2. Daraja la maneno katika x squared y squared minus 16 ni 4 na 0, na kiwango cha binomial nzima ni 4. Daraja la maneno katika 3n cubed minus 9n mraba ni 3 na 2, na kiwango cha binomial nzima ni 3. Mstari wa sita huitwa “Trinomial,” na kila kiini katika mstari huu kina trinomial tofauti. Mstari wa saba huitwa “Shahada ya kila neno,” na kila kiini kina digrii za maneno matatu katika trinomial juu yake. Mstari wa nane huitwa “Shahada ya polynomial,” na kila kiini kina kiwango cha trinomial kwa ujumla. Daraja la maneno katika x squared minus 7x pamoja na 12 ni 2, 1, na 0, na kiwango cha trinomial nzima ni 2. Daraja la maneno katika 9a mraba pamoja na 6ab pamoja na b squared ni 2, 2, na 2, na kiwango cha trinomial kwa ujumla ni 2. Daraja la maneno katika 6m hadi nguvu ya nne minus m cubed n squared pamoja 8mn kwa nguvu ya tano ni 4, 5, na 6, na kiwango cha trinomial nzima ni 6. Daraja la maneno katika z hadi nguvu ya nne pamoja na 3z mraba minus 1 ni 4, 2, na 0, na kiwango cha trinomial nzima ni 4. Mstari wa tisa huitwa “Polynomial,” na kila kiini kina polynomial tofauti. Mstari wa kumi huitwa “Shahada ya kila neno,” na safu ya kumi na moja inaitwa “Shahada ya polynomial.” Daraja la maneno katika b plus 1 ni 1 na 0, na kiwango cha polynomial nzima ni 1. Daraja la maneno katika 4y squared minus 7y pamoja na 2 ni 2, 1, na 0, na kiwango cha polynomial nzima ni 2. Daraja la maneno katika 4x hadi nguvu ya nne pamoja na x cubed pamoja na 8x mraba bala 9x pamoja 1 ni 4, 3, 2, 1, na 0, na kiwango cha polynomial nzima ni 4.

    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.

    Mfano\(\PageIndex{4}\)

    Pata shahada ya polynomials zifuatazo.

    1. 10y
    2. \(4 x^{3}-7 x+5\)
    3. -15
    4. \(-8 b^{2}+9 b-2\)
    5. \(8 x y^{2}+2 y\)
    Jibu
    1. \(\begin{array}{ll} & 10y\\ \text{The exponent of y is one. } y=y^1 & \text{The degree is 1.}\end{array}\)
    2. \(\begin{array}{ll} & 4 x^{3}-7 x+5\\ \text{The highest degree of all the terms is 3.} & \text{The degree is 3.}\end{array}\)
    3. \(\begin{array}{ll} & -15\\ \text{The degree of a constant is 0.} & \text{The degree is 0.}\end{array}\)
    4. \(\begin{array}{ll} & -8 b^{2}+9 b-2\\ \text{The highest degree of all the terms is 2.} & \text{The degree is 2.}\end{array}\)
    5. \(\begin{array}{ll} & 8 x y^{2}+2 y\\ \text{The highest degree of all the terms is 3.} & \text{The degree is 3.}\end{array}\)
    Mfano\(\PageIndex{5}\)

    Pata shahada ya polynomials zifuatazo:

    1. -15b
    2. \(10 z^{4}+4 z^{2}-5\)
    3. \(12 c^{5} d^{4}+9 c^{3} d^{9}-7\)
    4. \(3 x^{2} y-4 x\)
    5. —9
    Jibu
    1. 1
    2. 4
    3. 12
    4. 3
    5. 0
    Mfano\(\PageIndex{6}\)

    Pata shahada ya polynomials zifuatazo:

    1. 52
    2. \(a^{4} b-17 a^{4}\)
    3. \(5 x+6 y+2 z\)
    4. \(3 x^{2}-5 x+7\)
    5. \(-a^{3}\)
    Jibu
    1. 0
    2. 5
    3. 1
    4. 2
    5. 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.

    Mfano\(\PageIndex{7}\)

    Ongeza:\(25 y^{2}+15 y^{2}\)

    Jibu

    \(\begin{array}{ll} & 25 y^{2}+15 y^{2}\\ \text{Combine like terms.} & 40y^{2}\end{array}\)

    Mfano\(\PageIndex{8}\)

    Ongeza:\(12 q^{2}+9 q^{2}\)

    Jibu

    21\(q^{2}\)

    Mfano\(\PageIndex{9}\)

    Ongeza:\(-15 c^{2}+8 c^{2}\)

    Jibu

    \(-7 c^{2}\)

    Mfano\(\PageIndex{10}\)

    Ondoa: 16p- (-7p)

    Jibu

    \(\begin{array}{ll} & 16p−(−7p) \\ \text{Combine like terms.} & 23p\end{array}\)

    Mfano\(\PageIndex{11}\)

    Ondoa: 8m-m (-5m).

    Jibu

    13m

    Mfano\(\PageIndex{12}\)

    Ondoa:\(-15 z^{3}-\left(-5 z^{3}\right)\)

    Jibu

    \(-10 z^{3}\)

    Kumbuka kwamba kama maneno lazima kuwa na vigezo sawa na exponents sawa.

    Mfano\(\PageIndex{13}\)

    Kurahisisha:\(c^{2}+7 d^{2}-6 c^{2}\)

    Jibu

    \(\begin{array}{ll} & c^{2}+7 d^{2}-6 c^{2} \\ \text{Combine like terms.} & -5 c^{2}+7 d^{2} \end{array}\)

    Mfano\(\PageIndex{14}\)

    Ongeza:\(8 y^{2}+3 z^{2}-3 y^{2}\)

    Jibu

    \(5 y^{2}+3 z^{2}\)

    Mfano\(\PageIndex{15}\)

    Ongeza:\(3 m^{2}+n^{2}-7 m^{2}\)

    Jibu

    \(-4 m^{2}+n^{2}\)

    Mfano\(\PageIndex{16}\)

    Kurahisisha:\(u^{2} v+5 u^{2}-3 v^{2}\)

    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}\)

    Mfano\(\PageIndex{17}\)

    Kurahisisha:\(m^{2} n^{2}-8 m^{2}+4 n^{2}\)

    Jibu

    Hakuna maneno kama hayo ya kuchanganya.

    Mfano\(\PageIndex{18}\)

    Kurahisisha:\(p q^{2}-6 p-5 q^{2}\)

    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.

    Mfano\(\PageIndex{19}\)

    Pata jumla:\(\left(5 y^{2}-3 y+15\right)+\left(3 y^{2}-4 y-11\right)\)

    Jibu
    Tambua maneno kama hayo. 5 y mraba minus 3 y pamoja na 15, pamoja na 3 y squared minus 4 y minus 11.
    Panga upya ili kupata maneno kama hayo pamoja. 5y squared pamoja na 3y squared, kutambuliwa kama maneno, minus 3y minus 4y, kutambuliwa kama maneno, pamoja na 15 minus 11, kutambuliwa kama maneno.
    Kuchanganya kama maneno. 8 y mraba minus 7y pamoja na 4.
    Mfano\(\PageIndex{20}\)

    Pata jumla:\(\left(7 x^{2}-4 x+5\right)+\left(x^{2}-7 x+3\right)\)

    Jibu

    \(8 x^{2}-11 x+1\)

    Mfano\(\PageIndex{21}\)

    Pata jumla:\(\left(14 y^{2}+6 y-4\right)+\left(3 y^{2}+8 y+5\right)\)

    Jibu

    \(17 y^{2}+14 y+1\)

    Mfano\(\PageIndex{22}\)

    Pata tofauti:\(\left(9 w^{2}-7 w+5\right)-\left(2 w^{2}-4\right)\)

    Jibu
      9 w mraba minus 7 w pamoja na 5, minus 2 w squared minus 4.
    Kusambaza na kutambua maneno kama hayo. 9 w squared na 2 w squared ni kama maneno. 5 na 4 pia ni kama maneno.
    Panga upya masharti. 9 w squared minus 2 w squared minus 7 w pamoja 5 pamoja na 4.
    Kuchanganya kama maneno. 7 w mraba minus 7 w pamoja na 9.
    Mfano\(\PageIndex{23}\)

    Pata tofauti:\(\left(8 x^{2}+3 x-19\right)-\left(7 x^{2}-14\right)\)

    Jibu

    \(15 x^{2}+3 x-5\)

    Mfano\(\PageIndex{24}\)

    Pata tofauti:\(\left(9 b^{2}-5 b-4\right)-\left(3 b^{2}-5 b-7\right)\)

    Jibu

    \(6 b^{2}+3\)

    Mfano\(\PageIndex{25}\)

    Ondoa:\(\left(c^{2}-4 c+7\right)\) kutoka\(\left(7 c^{2}-5 c+3\right)\)

    Jibu
      .
      7 c mraba minus 5 c pamoja na 3, minus c squared minus 4c pamoja 7.
    Kusambaza na kutambua maneno kama hayo. 7 c squared na c squared ni kama maneno. Minus 5c na 4c ni kama maneno. 3 na minus 7 ni kama maneno.
    Panga upya masharti. 7 c mraba minus c mraba minus 5 c pamoja na 4 c pamoja na 3 minus 7.
    Kuchanganya kama maneno. 6 c mraba minus c minus 4.
    Mfano\(\PageIndex{26}\)

    Ondoa:\(\left(5 z^{2}-6 z-2\right)\) kutoka\(\left(7 z^{2}+6 z-4\right)\)

    Jibu

    \(2 z^{2}+12 z-2\)

    Mfano\(\PageIndex{27}\)

    Ondoa:\(\left(x^{2}-5 x-8\right)\) kutoka\(\left(6 x^{2}+9 x-1\right)\)

    Jibu

    \(5 x^{2}+14 x+7\)

    Mfano\(\PageIndex{28}\)

    Pata jumla:\(\left(u^{2}-6 u v+5 v^{2}\right)+\left(3 u^{2}+2 u v\right)\)

    Jibu

    \(\begin{array} {ll} & {\left(u^{2}-6 u v+5 v^{2}\right)+\left(3 u^{2}+2 u v\right)} \\\text{Distribute.} & {u^{2}-6 u v+5 v^{2}+3 u^{2}+2 u v} \\ \text{Rearrange the terms, to put like terms together} & {u^{2}+3 u^{2}-6 u v+2 u v+5 v^{2}} \\ \text{Combine like terms.} & {4 u^{2}-4 u v+5 v^{2}}\end{array}\)

    Mfano\(\PageIndex{29}\)

    Pata jumla:\(\left(3 x^{2}-4 x y+5 y^{2}\right)+\left(2 x^{2}-x y\right)\)

    Jibu

    \(5 x^{2}-5 x y+5 y^{2}\)

    Mfano\(\PageIndex{30}\)

    Pata jumla:\(\left(2 x^{2}-3 x y-2 y^{2}\right)+\left(5 x^{2}-3 x y\right)\)

    Jibu

    \(7 x^{2}-6 x y-2 y^{2}\)

    Mfano\(\PageIndex{31}\)

    Pata tofauti:\(\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)\)

    Jibu

    \(\begin{array}{ll} & {\left(p^{2}+q^{2}\right)-\left(p^{2}+10 p q-2 q^{2}\right)} \\ \text{Distribute.} &{p^{2}+q^{2}-p^{2}-10 p q+2 q^{2}} \\\text{Rearrange the terms, to put like terms together} & {p^{2}-p^{2}-10 p q+q^{2}+2 q^{2}} \\\text{Combine like terms.} & {-10 p q+3 q^{2}}\end{array}\)

    Mfano\(\PageIndex{32}\)

    Pata tofauti:\(\left(a^{2}+b^{2}\right)-\left(a^{2}+5 a b-6 b^{2}\right)\)

    Jibu

    \(-5 a b-5 b^{2}\)

    Mfano\(\PageIndex{33}\)

    Pata tofauti:\(\left(m^{2}+n^{2}\right)-\left(m^{2}-7 m n-3 n^{2}\right)\)

    Jibu

    \(4 n^{2}+7 m n\)

    Mfano\(\PageIndex{34}\)

    Kurahisisha:\(\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)\)

    Jibu

    \(\begin{array}{ll } & {\left(a^{3}-a^{2} b\right)-\left(a b^{2}+b^{3}\right)+\left(a^{2} b+a b^{2}\right)} \\ \text{Distribute.} &{a^{3}-a^{2} b-a b^{2}-b^{3}+a^{2} b+a b^{2}} \\ \text{Rearrange the terms, to put like terms together} & {a^{3}-a^{2} b+a^{2} b-a b^{2}+a b^{2}-b^{3}} \\ \text{Combine like terms.} &{a^{3}-b^{3}}\end{array}\)

    Mfano\(\PageIndex{35}\)

    Kurahisisha:\(\left(x^{3}-x^{2} y\right)-\left(x y^{2}+y^{3}\right)+\left(x^{2} y+x y^{2}\right)\)

    Jibu

    \(x^{3}-y^{3}\)

    Mfano\(\PageIndex{36}\)

    Kurahisisha:\(\left(p^{3}-p^{2} q\right)+\left(p q^{2}+q^{3}\right)-\left(p^{2} q+p q^{2}\right)\)

    Jibu

    \(p^{3}-2 p^{2} q+q^{3}\)

    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.

    Mfano\(\PageIndex{37}\)

    Tathmini\(5x^{2}−8x+4\) wakati

    1. x=4
    2. x=—2
    3. x=0
    Jibu
    1. x=4  
      5 x mraba minus 8 x pamoja 4.
    mbadala 4 kwa x. Mara 5 4 mraba minus mara 8 4 pamoja na 4.
    Kurahisisha watetezi. Mara 5 16 minus mara 8 4 pamoja na 4.
    Kuzidisha. 80 minus 32 pamoja na 4.
    Kurahisisha. 52.
    2. x=-1  
      5 x mraba minus 8 x pamoja 4.
    Mbadala hasi 2 kwa x. Mara 5 hasi 2 mraba minus mara 8 hasi 2 pamoja na 4.
    Kurahisisha watetezi. Mara 5 4 minus mara 8 hasi 2 pamoja na 4.
    Kuzidisha. 20 pamoja 16 pamoja 4.
    Kurahisisha. 40.
    3. x=0  
      5 x mraba minus 8 x pamoja 4.
    mbadala 0 kwa x. Mara 5 0 mraba minus mara 8 0 pamoja na 4.
    Kurahisisha watetezi. Mara 5 0 chini ya mara 8 0 pamoja na 4.
    Kuzidisha. 0 pamoja 0 pamoja 4.
    Kurahisisha. 4.
    Mfano\(\PageIndex{38}\)

    Tathmini:\(3x^{2}+2x−15\) wakati

    1. x=3
    2. x=-5
    3. x=0
    Jibu
    1. 18
    2. 50
    3. -15
    Mfano\(\PageIndex{39}\)

    Tathmini:\(5z^{2}−z−4\) wakati

    1. z=—2
    2. z=0
    3. z=2
    Jibu
    1. 18
    2. —4
    3. 14
    Mfano\(\PageIndex{40}\)

    Polynomial\(−16t^{2}+250\) anatoa urefu wa mpira tt sekunde baada ya imeshuka kutoka 250 mguu mrefu jengo. Pata urefu baada ya sekunde t=2.

    Jibu

    \(\begin{array}{ll } & −16t^{2}+250 \\ \text{Substitute t = 2.} & -16(2)^{2} + 250 \\ \text{Simplify }& −16\cdot 4+250 \\ \text{Simplify }& -64 + 250\\ \text{Simplify }& 186 \\& \text{After 2 seconds the height of the ball is 186 feet. } \end{array}\)

    Mfano\(\PageIndex{41}\)

    Polynomial\(−16t^{2}+250\) anatoa urefu wa mpira tt sekunde baada ya imeshuka kutoka 250 mguu mrefu jengo. Pata urefu baada ya sekunde t=0.

    Jibu

    250

    Mfano\(\PageIndex{42}\)

    Polynomial\(−16t^{2}+250\) inatoa urefu wa mpira tt sekunde baada ya imeshuka kutoka 250 mguu mrefu jengo. Pata urefu baada ya sekunde t=3.

    Jibu

    106

    Mfano\(\PageIndex{43}\)

    Polynomial\(6x^{2}+15xy\) 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
      6 x mraba pamoja na 15 x y.
    Mbadala x sawa 4 na y sawa 6. Mara 6 4 mraba pamoja na mara 15 mara 4 6.
    Kurahisisha. 6 mara 16 pamoja na mara 15 mara 4 6.
    Kurahisisha. 96 pamoja 360.
    Kurahisisha. 456.
      Gharama ya kuzalisha sanduku ni $456.
    Mfano\(\PageIndex{43}\)

    Polynomial\(6x^{2}+15xy\) 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

    Mfano\(\PageIndex{44}\)

    Polynomial\(6x^{2}+15xy\) 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 fomu\(ax^{m}\), 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 fomu\(ax^m\), 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.