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7.3: Ongeza na Ondoa Maneno ya busara

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    Muhtasari

    Mwishoni mwa sehemu hii, utaweza:

    • Ongeza na uondoe maneno ya busara na denominator ya kawaida
    • Ongeza na uondoe maneno ya busara ambao denominators ni kinyume
    • Find denominator angalau ya kawaida ya maneno ya busara
    • Ongeza na uondoe maneno ya busara na denominators tofauti
    • Ongeza na uondoe kazi za busara

    Kabla ya kuanza, fanya jaribio hili la utayari.

    1. Ongeza:\(\dfrac{7}{10}+\dfrac{8}{15}\).
      Ikiwa umekosa tatizo hili, tathmini [kiungo].
    2. Ondoa:\(\dfrac{3x}{4}−\dfrac{8}{9}\).
      Ikiwa umekosa tatizo hili, tathmini [kiungo].
    3. Ondoa:\(6(2x+1)−4(x−5)\).
      Ikiwa umekosa tatizo hili, tathmini [kiungo].

    Ongeza na Ondoa Maneno ya busara na Denominator ya kawaida

    Ni hatua gani ya kwanza unayochukua unapoongeza sehemu ndogo za namba? Unaangalia ikiwa wana denominator ya kawaida. Ikiwa wanafanya, unaongeza nambari na uweke jumla juu ya denominator ya kawaida. Ikiwa hawana denominator ya kawaida, unapata moja kabla ya kuongeza.

    Ni sawa na maneno ya busara. Ili kuongeza maneno ya busara, lazima wawe na denominator ya kawaida. Wakati denominators ni sawa, unaongeza nambari na kuweka jumla juu ya denominator ya kawaida.

    MANUFAA YA KUJIELEZA NA KUONDOA

    Kama\(p\)\(q\), na\(r\) ni polynomials ambapo\(r\neq 0\), basi

    \[\dfrac{p}{r}+\dfrac{q}{r}=\dfrac{p+q}{r} \quad \text{and} \quad \dfrac{p}{r}−\dfrac{q}{r}=\dfrac{p−q}{r}\nonumber\]

    Ili kuongeza au kuondoa maneno ya busara na denominator ya kawaida, ongeza au uondoe namba na uweke matokeo juu ya denominator ya kawaida.

    Sisi daima kurahisisha maneno ya busara. Hakikisha kuzingatia, ikiwa inawezekana, baada ya kuondoa nambari ili uweze kutambua mambo yoyote ya kawaida.

    Kumbuka, pia, hatuwezi kuruhusu maadili ambayo ingeweza kufanya denominator sifuri. Ni thamani gani ya\(x\) inapaswa kutengwa katika mfano unaofuata?

    Mfano\(\PageIndex{1}\)

    Ongeza:\(\dfrac{11x+28}{x+4}+\dfrac{x^2}{x+4}\).

    Suluhisho

    Tangu denominator ni\(x+4\), ni lazima kuwatenga thamani\(x=−4\).

    \(\begin{array} {ll} &\dfrac{11x+28}{x+4}+\dfrac{x^2}{x+4},\space x\neq −4 \\ \begin{array} {l} \text{The fractions have a common denominator,} \\ \text{so add the numerators and place the sum} \\ \text{over the common denominator.} \end{array} &\dfrac{11x+28+x^2}{x+4} \\ & \\ \text{Write the degrees in descending order.} &\dfrac{x^2+11x+28}{x+4} \\ & \\ \text{Factor the numerator.} &\dfrac{(x+4)(x+7)}{x+4} \\ & \\ \text{Simplify by removing common factors.} &\dfrac{\cancel{(x+4)}(x+7)}{\cancel{x+4}} \\ & \\ \text{Simplify.} &x+7 \end{array}\)

    Maneno haya yanapunguza\(x+7\) lakini usemi wa awali ulikuwa na denominator ya\(x+4\) hivyo\(x\neq −4\).

    Jaribu\(\PageIndex{2}\)

    Kurahisisha:\(\dfrac{9x+14}{x+7}+\dfrac{x^2}{x+7}\).

    Jibu

    \(x+2\)

    Jaribu\(\PageIndex{3}\)

    Kurahisisha:\(\dfrac{x^2+8x}{x+5}+\dfrac{15}{x+5}\).

    Jibu

    \(x+3\)

    Ili kuondoa maneno ya busara, lazima pia wawe na denominator ya kawaida. Wakati denominators ni sawa, wewe Ondoa nambari na kuweka tofauti juu ya denominator ya kawaida. Kuwa makini na ishara unapoondoa binomial au trinomial.

    Mfano\(\PageIndex{4}\)

    Ondoa:\(\dfrac{5x^2−7x+3}{x^2−3x+18}−\dfrac{4x^2+x−9}{x^2−3x+18}\).

    Suluhisho

    \(\begin{array} {ll} &\dfrac{5x^2−7x+3}{x^2−3x+18}−\dfrac{4x^2+x−9}{x^2−3x+18} \\ & \\ \begin{array} {l} \text{Subtract the numerators and place the} \\ \text{difference over the common denominator.} \end{array} &\dfrac{5x^2−7x+3−(4x^2+x−9)}{x^2−3x+18} \\ & \\ \text{Distribute the sign in the numerator.} &\dfrac{5x^2−7x+3−4x^2−x+9}{x^2−3x−18} \\ & \\ \text{Combine like terms.} &\dfrac{x^2−8x+12}{x^2−3x−18} \\ & \\ \text{Factor the numerator and the denominator.} &\dfrac{(x−2)(x−6)}{(x+3)(x−6)} \\ & \\ \text{Simplify by removing common factors.} &\dfrac{(x−2)\cancel{(x−6)}}{(x+3)\cancel{(x−6)}} \\ & \\ &(x−2)(x+3) \end{array}\)

    Jaribu\(\PageIndex{5}\)

    Ondoa:\(\dfrac{4x^2−11x+8}{x^2−3x+2}−\dfrac{3x^2+x−3}{x^2−3x+2}\).

    Jibu

    \(\dfrac{x−11}{x−2}\)

    Jaribu\(\PageIndex{6}\)

    Ondoa:\(\dfrac{6x^2−x+20}{x^2−81}−\dfrac{5x^2+11x−7}{x^2−81}\).

    Jibu

    \(\dfrac{x−3}{x+9}\)

    Kuongeza na Ondoa Maneno ya busara ambao Denominators ni kinyume

    Wakati denominators ya maneno mawili ya busara ni kinyume, ni rahisi kupata denominator ya kawaida. Tunapaswa kuzidisha moja ya sehemu ndogo na\(\dfrac{−1}{−1}\).

    Hebu tuone jinsi hii inavyofanya kazi.

      .
    Panua sehemu ya pili na\(\dfrac{−1}{−1}\). .
    Denominators ni sawa. .
    Kurahisisha. .

    Kuwa makini na ishara unapofanya kazi na kupinga wakati sehemu ndogo zinaondolewa.

    Mfano\(\PageIndex{7}\)

    Ondoa:\(\dfrac{m^2−6m}{m^2−1}−\dfrac{3m+2}{1−m^2}\).

    Suluhisho

      .

    Denominators ni kinyume, hivyo kuzidisha sehemu ya
    pili na\(\dfrac{−1}{−1}\).

    		 .
    Kurahisisha sehemu ya pili. .
    Denominators ni sawa. Ondoa nambari. .
    Kusambaza. .
    Kuchanganya kama maneno. .
    Fanya namba na denominator. .
    Kurahisisha kwa kuondoa mambo ya kawaida. .
    Kurahisisha. .
    Jaribu\(\PageIndex{8}\)

    Ondoa:\(\dfrac{y^2−5y}{y^2−4}−\dfrac{6y−6}{4−y^2}\).

    Jibu

    \(\dfrac{y+3}{y+2}\)

    Jaribu\(\PageIndex{9}\)

    Ondoa:\(\dfrac{2n^2+8n−1}{n^2−1}−\dfrac{n^2−7n−1}{1−n^2}\).

    Jibu

    \(\dfrac{3n−2}{n−1}\)

    Pata Denominator ya kawaida ya maneno ya busara

    Tunapoongeza au kuondoa maneno ya busara na denominators tofauti, tutahitaji kupata denominators ya kawaida. Ikiwa tunapitia utaratibu tuliotumia kwa sehemu ndogo za namba, tutajua nini cha kufanya na maneno ya busara.

    Hebu tuangalie mfano huu:\(\dfrac{7}{12}+\dfrac{5}{18}\). Kwa kuwa denominators si sawa, hatua ya kwanza ilikuwa kupata denominator ya kawaida (LCD).

    Ili kupata LCD ya sehemu ndogo, tulifanya 12 na 18 katika primes, tukiweka primes yoyote ya kawaida katika nguzo. Kisha sisi “tulileta” mkuu mmoja kutoka kila safu. Hatimaye, tuliongeza sababu za kupata LCD.

    Tunapoongeza vipande vya namba, mara tulipopata LCD, tumeandika upya kila sehemu kama sehemu sawa na LCD kwa kuzidisha namba na denominator kwa idadi sawa. Sasa tuko tayari kuongeza.

    Saba-kumi na mbili pamoja na tano na nane. Andika factorizations mkuu wa kila denominator na line up mambo ya kawaida. Denominator ya sehemu ya kwanza ni 12. Factorization mkuu wa 12 ni mara 2 mara 3. Denominator ya sehemu ya pili ni 18. Factorization mkuu wa 18 ni mara 2 mara 3 3. Kuleta sababu kutoka kila safu, denominator ya kawaida ya 12 na 18 ni mara 2 mara 3 mara 3, ambayo ni 36. Andika sehemu zote mbili kwa kutumia denominator ya chini kabisa. Ili kufanya hivyo, kuzidisha nambari na denominator ya sehemu ya kwanza na 3 na kuzidisha namba na denominator ya sehemu ya pili na 2. Matokeo yake ni mara 7 3 yote imegawanywa na mara 12 3 pamoja na mara 5 2 yote imegawanywa na mara 18 2. Kurahisisha kila sehemu. 7 mara 3 ni 21 na 12 mara 3 ni 36. 5 mara 2 ni 10 na 18 mara 2 ni 36. Matokeo yake ni ishirini na moja thelathini na sita pamoja na kumi thelathini na sita.

    Tunafanya kitu kimoja kwa maneno ya busara. Hata hivyo, sisi kuondoka LCD katika fomu factored.

    PATA DENOMINATOR YA KAWAIDA YA MANENO YA BUSARA.
    1. Factor kila denominator kabisa.
    2. Orodha ya mambo ya kila denominator. Mechi sababu wima ikiwezekana.
    3. Kuleta nguzo kwa kuingiza mambo yote, lakini usijumuishe mambo ya kawaida mara mbili.
    4. Andika LCD kama bidhaa ya mambo.

    Kumbuka, sisi daima hutenganisha maadili ambayo yangeweza kufanya sifuri ya denominator. Ni maadili gani ya xx tunapaswa kuwatenga katika mfano huu unaofuata?

    Mfano\(\PageIndex{10}\)

    Pata LCD kwa maneno\(\dfrac{8}{x^2−2x−3}\),\(\dfrac{3x}{x^2+4x+3}\) na b. kuandika tena kama maneno sawa ya busara na denominator ya chini ya kawaida.

    Suluhisho

    a.

    Kupata LCD kwa\(\dfrac{8}{x^2−2x−3}\),\(\dfrac{3x}{x^2+4x+3}\).  
    Sababu kila denominator kabisa, kuunganisha mambo ya kawaida.

    Kuleta chini nguzo.    		 .
    Andika LCD kama bidhaa ya mambo. .

    b.

      .
    Factor kila denominator. .
    Kuzidisha kila denominator kwa sababu ya 'kukosa'
    LCD na kuzidisha kila nambari kwa sababu sawa.    		 .
    Kurahisisha nambari. .
    Jaribu\(\PageIndex{11}\)

    Pata LCD kwa maneno\(\dfrac{2}{x^2−x−12}\), b. kuandika tena\(\dfrac{1}{x^2−16}\) kama maneno sawa ya busara na denominator ya chini ya kawaida.

    Jibu

    a.\((x−4)(x+3)(x+4)\)
    b.\(\dfrac{2x+8}{(x−4)(x+3)(x+4)}\),
    \(\dfrac{x+3}{(x−4)(x+3)(x+4)}\)

    Jaribu\(\PageIndex{12}\)

    Pata LCD kwa maneno\(\dfrac{3x}{x^2−3x+10}\), b. kuandika tena\(\dfrac{5}{x^2+3x+2}\) kama maneno sawa ya busara na denominator ya chini ya kawaida.

    Jibu

    a.\((x+2)(x−5)(x+1)\)
    b.\(\dfrac{3x^2+3x}{(x+2)(x−5)(x+1)}\),
    \(\dfrac{5x−25}{(x+2)(x−5)(x+1)}\)

    Kuongeza na Ondoa Maneno ya busara na Tofauti na Denominators

    Sasa tuna hatua zote tunayohitaji kuongeza au kuondoa maneno ya busara na tofauti na denominators.

    Mfano\(\PageIndex{13}\): How to Add Rational Expressions with Unlike Denominators

    Ongeza:\(\dfrac{3}{x−3}+\dfrac{2}{x−2}\).

    Suluhisho

    Hatua ya 1 ni kuamua kama maneno ya busara 3 yamegawanywa na wingi x minus 3 na 2 imegawanywa na wingi x minus 2 na mambo ya kawaida. Denominators x minus 3 na x minus 2 hawana sababu yoyote ya kawaida, ambayo ina maana denominator ya kawaida ya maneno ya busara ni kiasi x bala mara 3 kiasi x minus 2. Andika upya kila kujieleza kwa busara na denominator ya kawaida. Panua nambari na denominator ya 3 imegawanywa na wingi x minus 3 kwa kiasi x minus 2. Panua nambari na denominator ya 2 iliyogawanywa na kiasi x chini ya 2 kwa kiasi x chini ya 2. Matokeo yake ni kujieleza kwa busara mara 3 kiasi x minus 2 yote imegawanywa na wingi x bala mara 3 kiasi x minus 2 pamoja kujieleza busara mara 2 kiasi x minus 3 kugawanywa na wingi x bala mara 2 kiasi x minus 3. Kurahisisha nambari na kuweka denominators factored. Nambari ya kujieleza kwa busara ya kwanza, mara 3 kiasi x chini ya 2, inafungua hadi 3 x chini ya 6. Nambari ya kujieleza kwa busara ya pili, mara 2 kiasi x chini ya 3, inafungua hadi 2 x chini ya 6. Matokeo yake ni kujieleza kwa busara kiasi cha 3 x minus 6 yote imegawanywa na wingi x bala mara 3 kiasi x minus 2 pamoja na kujieleza busara, wingi 2 x minus 6 wote kugawanywa na wingi x bala mara 3 kiasi x minus 2.Hatua ya 2 ni kuongeza au kuondoa maneno ya busara kwa kuongeza namba, kiasi cha 3 x minus 6 na wingi 2 x minus 6, na kuweka jumla juu ya denominator, kiasi x bala mara 3 kiasi x bala 2. Matokeo yake ni wingi 3 x minus 6 pamoja na 2 x minus 6 wote kugawanywa na wingi x minus mara 3 kiasi x minus 2. Kurahisisha nambari kwa kuchanganya maneno kama hayo. Matokeo yake ni kiasi 5 x chini ya 12 yote imegawanywa na wingi x minus mara 3 kiasi x minus 2.Hatua ya 3. Angalia kwamba 5 x minus 12 haiwezi kuhesabiwa, hivyo jibu ni rahisi.

    Jaribu\(\PageIndex{14}\)

    Ongeza:\(\dfrac{2}{x−2}+\dfrac{5}{x+3}\).

    Jibu

    \(\dfrac{7x−4}{(x−2)(x+3)}\)

    Jaribu\(\PageIndex{15}\)

    Ongeza:\(\dfrac{4}{m+3}+\dfrac{3}{m+4}\).

    Jibu

    \(\dfrac{7m+25}{(m+3)(m+4)}\)

    Hatua zilizotumiwa kuongeza maneno ya busara zinafupishwa hapa.

    ONGEZA AU ONDOA MANENO YA BUSARA.
    1. Kuamua kama maneno yana denominator ya kawaida.
      • Ndiyo — nenda hatua ya 2.
      • Hapana — Andika upya kila kujieleza mantiki na LCD.
        • Kupata LCD.
        • Andika upya kila kujieleza kwa busara kama kujieleza sawa na LCD.
    2. Ongeza au uondoe maneno ya busara.
    3. Kurahisisha, ikiwa inawezekana.

    Epuka majaribu ya kurahisisha hivi karibuni. Katika mfano hapo juu, ni lazima tuondoke maneno ya kwanza ya busara ili tuweze kuiongezea\(\dfrac{2x−6}{(x−2)(x−3)}\).\(\dfrac{3x−6}{(x−3)(x−2)}\) Kurahisisha tu baada ya kuunganisha nambari.

    Mfano\(\PageIndex{16}\)

    Ongeza:\(\dfrac{8}{x^2−2x−3}+\dfrac{3x}{x^2+4x+3}\).

    Suluhisho

      .
    Je, maneno yana denominator ya kawaida? Hapana.
    Andika upya kila kujieleza na LCD.  
    \(\begin{array} {ll} & \\ & \\ \text{Find the LCD.} &\begin{array} {l} \hspace{5mm} x^2−2x−3=(x+1)(x−3) \\ \underline{x^2+4x+3=(x+1)\quad (x+3)} \\ & \\ \qquad LCD=(x+1)(x−3)(x+3) \end{array} \end{array} \)  
    Andika upya kila kujieleza kwa busara kama kujieleza
    sawa na LCD.    		 .
    Kurahisisha nambari. .
    Ongeza maneno ya busara. .
    Kurahisisha nambari. .
      Nambari ni mkuu, kwa hiyo
    hakuna sababu za kawaida.
    Jaribu\(\PageIndex{17}\)

    Ongeza:\(\dfrac{1}{m^2−m−2}+\dfrac{5m}{m^2+3m+2}\).

    Jibu

    \(\dfrac{5m^2−9m+2}{(m+1)(m−2)(m+2)}\)

    Jaribu\(\PageIndex{18}\)

    Ongeza:\(\dfrac{2n}{n^2−3n−10}+\dfrac{6}{n^2+5n+6}\).

    Jibu

    \(\dfrac{2n^2+12n−30}{(n+2)(n−5)(n+3)}\)

    Mchakato tunayotumia kuondoa maneno ya busara na denominators tofauti ni sawa na kwa kuongeza. Tunapaswa tu kuwa makini sana kwa ishara wakati wa kuondoa nambari.

    Mfano\(\PageIndex{19}\)

    Ondoa:\(\dfrac{8y}{y^2−16}−\dfrac{4}{y−4}\).

    Suluhisho

      .
    Je, maneno yana denominator ya kawaida? Hapana.
    Andika upya kila kujieleza na LCD.  
    \(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} y^2−16=(y−4)(y+4) \\ \quad \underline{y−4=y−4} \\ LCD=(y−4)(y+4) \end{array} \end{array} \)  
    Andika upya kila kujieleza kwa busara kama kujieleza
    sawa na LCD.    		 .
    Kurahisisha nambari. .
    Ondoa maneno ya busara. .
    Kurahisisha nambari. .
    Fanya namba ya kuangalia mambo ya kawaida. .
    Ondoa mambo ya kawaida .
    Kurahisisha. .
    Jaribu\(\PageIndex{20}\)

    Ondoa:\(\dfrac{2x}{x^2−4}−\dfrac{1}{x+2}\).

    Jibu

    \(\dfrac{1}{x−2}\)

    Jaribu\(\PageIndex{21}\)

    Ondoa:\(\dfrac{3}{z+3}−\dfrac{6z}{z^2−9}\).

    Jibu

    \(\dfrac{−3}{z−3}\)

    Kuna ishara nyingi hasi katika mfano unaofuata. Kuwa makini zaidi.

    Mfano\(\PageIndex{22}\)

    Ondoa:\(\dfrac{−3n−9}{n^2+n−6}−\dfrac{n+3}{2−n}\).

    Suluhisho

      .
    Sababu ya denominator. .
    Tangu\(n−2\) na\(2−n\) ni kinyume,
    tutazidisha maneno ya pili ya busara na\(\dfrac{−1}{−1}\).    		 .
    . .
    Kurahisisha. Kumbuka,\(a−(−b)=a+b\). .
    Je! Maneno ya busara yana denominator ya
    kawaida? Hapana.    		  
    \(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} n^2+n−6=(n−2)(n+3) \\ \quad\underline{n−2=(n−2)} \\ LCD=\quad (n−2)(n+3) \end{array} \end{array} \)  
    Andika upya kila kujieleza kwa busara kama kujieleza
    sawa na LCD.    		 .
    Kurahisisha nambari. .
    Ongeza maneno ya busara. .
    Kurahisisha nambari. .
    Fanya namba ya kuangalia mambo ya kawaida. .
    Kurahisisha. .
    Jaribu\(\PageIndex{23}\)

    Ondoa:\(\dfrac{3x−1}{x^2−5x−6}−\dfrac{2}{6−x}\).

    Jibu

    \(\dfrac{5x+1}{(x−6)(x+1)}\)

    Jaribu\(\PageIndex{24}\)

    Ondoa:\(\dfrac{−2y−2}{y^2+2y−8}−\dfrac{y−1}{2−y}\).

    Jibu

    \(\dfrac{y+3}{y+4}\)

    Mambo yanaweza kupata messy sana wakati sehemu zote mbili zinapaswa kuzidishwa na binomial ili kupata denominator ya kawaida.

    Mfano\(\PageIndex{25}\)

    Ondoa:\(\dfrac{4}{a^2+6a+5}−\dfrac{3}{a^2+7a+10}\).

    Suluhisho

      .
    Sababu ya denominators. .
    Je! Maneno ya busara yana denominator ya
    kawaida? Hapana.    		  

    \(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} a^2+6a+5=(a+1)(a+5) \\ \underline{a^2+7a+10=(a+5)(a+2)} \\ LCD=(a+1)(a+5)(a+2) \end{array} \end{array} \)

    		  
    Andika upya kila kujieleza kwa busara kama kujieleza
    sawa na LCD.    		 .
    Kurahisisha nambari. .
    Ondoa maneno ya busara. .
    Kurahisisha nambari. .
      .
    Angalia mambo ya kawaida. .
    Kurahisisha. .
    Jaribu\(\PageIndex{26}\)

    Ondoa:\(\dfrac{3}{b^2−4b−5}−\dfrac{2}{b^2−6b+5}\).

    Jibu

    \(\dfrac{1}{(b+1)(b−1)}\)

    Jaribu\(\PageIndex{27}\)

    Ondoa:\(\dfrac{4}{x^2−4}−\dfrac{3}{x^2−x−2}\).

    Jibu

    \(\dfrac{1}{(x+2)(x+1)}\)

    Sisi kufuata hatua sawa na kabla ya kupata LCD wakati tuna maneno zaidi ya mbili busara. Katika mfano unaofuata, tutaanza kwa kuzingatia madhehebu yote matatu ili kupata LCD yao.

    Mfano\(\PageIndex{28}\)

    Kurahisisha:\(\dfrac{2u}{u−1}+\dfrac{1}{u}−\dfrac{2u−1}{u^2−u}\).

    Suluhisho

      .
    Je, maneno yana denominator ya kawaida? Hapana.
    Andika upya kila kujieleza na LCD.    		  
    \(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} u−1=(u−1) \\ u=u \\ \underline{u^2−u=u(u−1)} \\ LCD=u(u−1) \end{array} \end{array}\)  
    Andika upya kila kujieleza kwa busara kama kujieleza
    sawa na LCD.    		 .
      .
    Andika kama kujieleza moja ya busara. .
    Kurahisisha. .
    Fanya namba, na uondoe mambo
    ya kawaida.    		 .
    Kurahisisha. .
    Jaribu\(\PageIndex{29}\)

    Kurahisisha:\(\dfrac{v}{v+1}+\dfrac{3}{v−1}−\dfrac{6}{v^2−1}\).

    Jibu

    \(\dfrac{v+3}{v+1}\)

    Jaribu\(\PageIndex{30}\)

    Kurahisisha:\(\dfrac{3w}{w+2}+\dfrac{2}{w+7}−\dfrac{17w+4}{w^2+9w+14}\).

    Jibu

    \(\dfrac{3w}{w+7}\)

    Ongeza na uondoe kazi za busara

    Ili kuongeza au kuondoa kazi za busara, tunatumia mbinu sawa ambazo tulizitumia kuongeza au kuondoa kazi nyingi.

    Mfano\(\PageIndex{31}\)

    Pata\(R(x)=f(x)−g(x)\) wapi\(f(x)=\dfrac{x+5}{x−2}\) na\(g(x)=\dfrac{5x+18}{x^2−4}\).

    ufumbuzi
      .
    Mbadala katika kazi\(f(x)\),\(g(x)\). .
    Sababu ya denominators. .
    Je, maneno yana denominator ya kawaida? Hapana.
    Andika upya kila kujieleza na LCD.    		  
    \(\begin{array} {ll} \text{Find the LCD.} &\begin{array} {l} x−2=(x−2) \\ \underline{x^2−4=(x−2)(x+2)} \\ \hspace{4mm} LCD=(x−2)(x+2)\end{array} \end{array}\)  
    Andika upya kila kujieleza kwa busara kama kujieleza
    sawa na LCD.    		 .
    Andika kama kujieleza moja ya busara. .
    Kurahisisha. .
      .
    Fanya namba, na uondoe mambo
    ya kawaida.    		 .
    Kurahisisha. .
    Jaribu\(\PageIndex{32}\)

    Pata\(R(x)=f(x)−g(x)\) wapi\(f(x)=\dfrac{x+1}{x+3}\) na\(g(x)=\dfrac{x+17}{x^2−x−12}\).

    Jibu

    \(\dfrac{x−7}{x−4}\)

    Jaribu\(\PageIndex{33}\)

    Pata\(R(x)=f(x)+g(x)\) wapi\(f(x)=\dfrac{x−4}{x+3}\) na\(g(x)=\dfrac{4x+6}{x^2−9}\).

    Jibu

    \(\dfrac{x^2−3x+18}{(x+3)(x−3)}\)

    Fikia rasilimali hii ya mtandaoni kwa maelekezo ya ziada na mazoezi kwa kuongeza na kuondoa maneno ya busara.

    • Kuongeza na Ondoa Maneno ya busara- Tofauti na Denominators

    Dhana muhimu

    • Mantiki kujieleza Aidha na Ondoa
      Kama\(p\),\(q\), na\(r\) ni polynomials ambapo\(r\neq 0\), basi
      \[\dfrac{p}{r}+\dfrac{q}{r}=\dfrac{p+q}{r} \quad \text{and} \quad \dfrac{p}{r}−\dfrac{q}{r}=\dfrac{p−q}{r}\nonumber\]
    • Jinsi ya kupata denominator ya kawaida ya maneno ya busara.
      1. Factor kila kujieleza kabisa.
      2. Orodha ya mambo ya kila kujieleza. Mechi sababu wima ikiwezekana.
      3. Kuleta chini nguzo.
      4. Andika LCD kama bidhaa ya mambo.
    • Jinsi ya kuongeza au kuondoa maneno ya busara.
      1. Kuamua kama maneno yana denominator ya kawaida.
        • Ndiyo — nenda hatua ya 2.
        • Hapana — Andika upya kila kujieleza mantiki na LCD.
          • Kupata LCD.
          • Andika upya kila kujieleza kwa busara kama kujieleza sawa na LCD.
      2. Ongeza au uondoe maneno ya busara.
      3. Kurahisisha, ikiwa inawezekana.