6.6: Gawanya Polynomials

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Gawanya polynomial na monomial
• Gawanya polynomial na binomial

Kumbuka

Kabla ya kuanza, fanya jaribio hili la utayari.

1. Kuongeza:$\dfrac{3}{d}+\dfrac{x}{d}$
   Kama amekosa tatizo hili, kupitia Zoezi 1.7.1.
2. kurahisisha:$\dfrac{30 x y^{3}}{5 x y}$
   Kama amekosa tatizo hili, mapitio Zoezi 6.5.37.
3. Kuchanganya kama maneno:$8 a^{2}+12 a+1+3 a^{2}-5 a+4$
   Kama amekosa tatizo hili, mapitio Zoezi 1.3.37.

Gawanya Polynomial na Monomial

Katika sehemu ya mwisho, umejifunza jinsi ya kugawanya monomial na monomial. Unapoendelea kujenga ujuzi wako wa polynomials utaratibu unaofuata ni kugawanya polynomial ya maneno mawili au zaidi na monomial.

Njia tutakayotumia kugawanya polynomial na monomial inategemea mali ya kuongeza sehemu. Hivyo tutaweza kuanza na mfano kupitia sehemu Aidha.

$\begin{array}{ll}{\text { The sum, }} & {\dfrac{y}{5}+\dfrac{2}{5}} \\ {\text { simplifies to }} & {\dfrac{y+2}{5}}\end{array}$

Sasa tutafanya hivyo kwa reverse ili kugawanya sehemu moja katika vipande tofauti.

Tutaweza hali sehemu ya kuongeza mali hapa kama wewe kujifunza na katika reverse.

SEHEMU YA KUONGEZA

Ikiwa a, b, na c ni namba ambapo$c\neq 0$, basi

$$\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c} \quad \text { and } \quad \dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}$$

Tunatumia fomu upande wa kushoto ili kuongeza sehemu ndogo na tunatumia fomu kwa haki ya kugawanya polynomial na monomial.

$\begin{array}{ll}{\text { For example, }} & {\dfrac{y+2}{5}} \\ {\text { can be written }} & {\dfrac{y}{5}+\dfrac{2}{5}}\end{array}$

Tunatumia fomu hii ya kuongeza sehemu ili kugawanya polynomials na monomials.

MGAWANYIKO WA POLYNOMIAL NA MONOMIAL

Ili kugawanya polynomial na monomial, kugawanya kila neno la polynomial na monomial.

Zoezi$\PageIndex{1}$

Pata quotient:$\dfrac{7 y^{2}+21}{7}$

Jibu

$\begin{array}{ll} & \dfrac{7 y^{2}+21}{7}\\\text{Divide each term of the numerator by the denominator.} & \dfrac{7 y^{2}}{7}+\dfrac{21}{7} \\ \text {Simplify each fraction. } & y^{2}+3 \end{array}$

Zoezi$\PageIndex{2}$

Pata quotient:$\dfrac{8 z^{2}+24}{4}$

Jibu

$2 z^{2}+6$

Zoezi$\PageIndex{3}$

Pata quotient:$\dfrac{18 z^{2}-27}{9}$

Jibu

$2 z^{2}-3$

Kumbuka kwamba mgawanyiko unaweza kuwakilishwa kama sehemu. Unapoulizwa kugawanya polynomial na monomial na si tayari katika fomu ya sehemu, andika sehemu na polynomial katika nambari na monomial katika denominator.

Zoezi$\PageIndex{4}$

Pata quotient:$\left(18 x^{3}-36 x^{2}\right) \div 6 x$

Jibu

$\begin{array}{ll} & \left(18 x^{3}-36 x^{2}\right) \div 6 x\\\text { Rewrite as a fraction. } & \dfrac{18 x^{3}-36 x^{2}}{6 x} \\ \text { Divide each term of the numerator by the denominator. }& \dfrac{18 x^{3}}{6 x}-\dfrac{36 x^{2}}{6 x}\\ \text { Simplify. } &3 x^{2}-6 x\end{array}$

Zoezi$\PageIndex{5}$

Pata quotient:$\left(27 b^{3}-33 b^{2}\right) \div 3 b$

Jibu

$9 b^{2}-11 b$

Zoezi$\PageIndex{6}$

Pata quotient:$\left(25 y^{3}-55 y^{2}\right) \div 5 y$

Jibu

$5 y^{2}-11 y$

Tunapogawanya na hasi, lazima tuwe makini zaidi na ishara.

Zoezi$\PageIndex{7}$

Pata quotient:$\dfrac{12 d^{2}-16 d}{-4}$

Jibu

$\begin{array}{ll} &\dfrac{12 d^{2}-16 d}{-4}\\ \text { Divide each term of the numerator by the denominator. }& \dfrac{18 x^{3}-36 x^{2}}{6 x} \\ \text { Simplify. Remember, subtracting a negative is like adding a positive! }& -3 d^{2}+4 d\end{array}$

Zoezi$\PageIndex{8}$

Pata quotient:$\dfrac{25 y^{2}-15 y}{-5}$

Jibu

$-5 y^{2}+3 y$

Zoezi$\PageIndex{9}$

Pata quotient:$\dfrac{42 b^{2}-18 b}{-6}$

Jibu

$-7 b^{2}+3 b$

Zoezi$\PageIndex{10}$

Pata quotient:$\dfrac{105 y^{5}+75 y^{3}}{5 y^{2}}$

Jibu

$\begin{array}{ll} &\dfrac{105 y^{5}+75 y^{3}}{5 y^{2}}\\ \text { Separate the terms. }& \dfrac{105 y^{5}}{5 y^{2}}+\dfrac{75 y^{3}}{5 y^{2}}\\ \text { Simplify. }& 21 y^{3}+15 y\end{array}$

Zoezi$\PageIndex{11}$

Pata quotient:$\dfrac{60 d^{7}+24 d^{5}}{4 d^{3}}$

Jibu

$15 d^{4}+6 d^{2}$

Zoezi$\PageIndex{12}$

Pata quotient:$\dfrac{216 p^{7}-48 p^{5}}{6 p^{3}}$

Jibu

$36 p^{4}-8 p^{2}$

Zoezi$\PageIndex{13}$

Pata quotient:$\left(15 x^{3} y-35 x y^{2}\right) \div(-5 x y)$

Jibu

$\begin{array}{ll} &\left(15 x^{3} y-35 x y^{2}\right) \div(-5 x y)\\ \text { Rewrite as a fraction. }& \dfrac{15 x^{3} y-35 x y^{2}}{-5 x y}\\\text { Separate the terms. Be careful with the signs! }& \dfrac{15 x^{3} y}{-5 x y}-\dfrac{35 x y^{2}}{-5 x y}\\ \text { Simplify. } & -3 x^{2}+7 y\end{array}$

Zoezi$\PageIndex{14}$

Pata quotient:$\left(32 a^{2} b-16 a b^{2}\right) \div(-8 a b)$

Jibu

$-4 a+2 b$

Zoezi$\PageIndex{15}$

Pata quotient:$\left(-48 a^{8} b^{4}-36 a^{6} b^{5}\right) \div\left(-6 a^{3} b^{3}\right)$

Jibu

$8 a^{5} b+6 a^{3} b^{2}$

Zoezi$\PageIndex{13}$

Pata quotient:$\dfrac{36 x^{3} y^{2}+27 x^{2} y^{2}-9 x^{2} y^{3}}{9 x^{2} y}$

Jibu

$\begin{array}{ll} &\dfrac{36 x^{3} y^{2}+27 x^{2} y^{2}-9 x^{2} y^{3}}{9 x^{2} y}\\\text { Separate the terms. }& \dfrac{36 x^{3} y^{2}}{9 x^{2} y}+\dfrac{27 x^{2} y^{2}}{9 x^{2} y}-\dfrac{9 x^{2} y^{3}}{9 x^{2} y}\\ \text { Simplify. } & 4 x y+3 y-y^{2}\end{array}$

Zoezi$\PageIndex{14}$

Pata quotient:$\dfrac{40 x^{3} y^{2}+24 x^{2} y^{2}-16 x^{2} y^{3}}{8 x^{2} y}$

Jibu

$5 x y+3 y-2 y^{2}$

Zoezi$\PageIndex{15}$

Pata quotient:$\dfrac{35 a^{4} b^{2}+14 a^{4} b^{3}-42 a^{2} b^{4}}{7 a^{2} b^{2}}$

Jibu

$5 a^{2}+2 a^{2} b-6 b^{2}$

Zoezi$\PageIndex{16}$

Pata quotient:$\dfrac{10 x^{2}+5 x-20}{5 x}$

Jibu

$\begin{array}{ll}&\dfrac{10 x^{2}+5 x-20}{5x}\\\text { Separate the terms. }& \dfrac{10 x^{2}}{5 x}+\dfrac{5 x}{5 x}-\dfrac{20}{5 x}\\ \text { Simplify. } &2 x+1-\dfrac{4}{x}\end{array}$

Zoezi$\PageIndex{17}$

Pata quotient:$\dfrac{18 c^{2}+6 c-9}{6 c}$

Jibu

$3 c+1-\dfrac{3}{2 c}$

Zoezi$\PageIndex{18}$

Pata quotient:$\dfrac{10 d^{2}-5 d-2}{5 d}$

Jibu

$2 d-1-\dfrac{2}{5 d}$

Gawanya Polynomial na Binomial

Ili kugawanya polynomial na binomial, tunafuata utaratibu sawa na mgawanyiko mrefu wa idadi. Basi hebu tuangalie kwa makini hatua tunazochukua wakati tunagawanya nambari ya tarakimu 3, 875, na namba ya tarakimu 2, 25.

Tunaandika mgawanyiko mrefu
Tunagawanya tarakimu mbili za kwanza, 87, na 25.
Tunazidisha mara 3 25 na kuandika bidhaa chini ya 87.
Sasa tunaondoa 75 kutoka 87.
Kisha sisi kuleta chini tarakimu ya tatu ya mgao, 5.
Kurudia mchakato, kugawanya 25 hadi 125.

Tunaangalia mgawanyiko kwa kuzidisha quotient na mgawanyiko.

Ikiwa tulifanya mgawanyiko kwa usahihi, bidhaa lazima iwe sawa na mgao.

$$\begin{array}{l}{35 \cdot 25} \\ {875}\checkmark\end{array}$$

Sasa tutagawanya trinomial na binomial. Unaposoma kupitia mfano, angalia jinsi hatua zilivyo sawa na mfano wa namba hapo juu.

Zoezi$\PageIndex{19}$

Pata quotient:$\left(x^{2}+9 x+20\right) \div(x+5)$

Jibu

Andika kama tatizo la mgawanyiko mrefu.
Kuwa na uhakika mgao ni katika hali ya kawaida.
Gawanya x ^2 na x. Inaweza kusaidia kujiuliza, “Ninahitaji nini kuzidisha x na kupata x ^2?”
Weka jibu, x, katika quotient juu ya muda x.
Kuzidisha x mara x + 5. Line up maneno kama chini ya mgao.
Ondoa x ^2 + 5 x kutoka x ^2 + 9 x.
Kisha kuleta chini ya muda wa mwisho, 20.
Gawanya 4 x na x. Inaweza kusaidia kujiuliza, “Ninahitaji nini kuzidisha x na kupata 4 x?”
Weka jibu, 4, katika quotient juu ya muda wa mara kwa mara.
Kuzidisha mara 4 x + 5.
Ondoa 4 x + 20 kutoka 4 x + 20.
Angalia:
Panua quotient na mgawanyiko.
(x + 4) (x + 5)
Unapaswa kupata mgao.
x ^2 + 9 x + 20 ✓

Zoezi$\PageIndex{20}$

Pata quotient:$\left(y^{2}+10 y+21\right) \div(y+3)$

Jibu

y+7

Zoezi$\PageIndex{21}$

Pata quotient:$\left(m^{2}+9 m+20\right) \div(m+4)$

Jibu

m+5

Wakati mgawanyiko ina Ondoa ishara, ni lazima kuwa makini zaidi wakati sisi kuzidisha quotient sehemu na kisha Ondoa. Inaweza kuwa salama kuonyesha kwamba tunabadilisha ishara na kisha kuongeza.

Zoezi$\PageIndex{22}$

Pata quotient:$\left(2 x^{2}-5 x-3\right) \div(x-3)$

Jibu

Andika kama tatizo la mgawanyiko mrefu.
Kuwa na uhakika mgao ni katika hali ya kawaida.
Gawanya 2 x ^2 na x. Weka jibu, 2 x, katika quotient juu ya muda x.
Kuzidisha 2 x mara x - 3. Line up maneno kama chini ya mgao.
Ondoa 2 x ^2 - 6 x kutoka 2 x ^2 - 5 x. Badilisha ishara na kisha uongeze. Kisha kuleta chini muda wa mwisho.
Gawanya x na x. Weka jibu, 1, katika quotient juu ya muda wa mara kwa mara.
Panua mara 1 x - 3.
Ondoa x - 3 kutoka x - 3 kwa kubadilisha ishara na kuongeza.
Kuangalia, kuzidisha (x - 3) (2 x + 1).
Matokeo yake yanapaswa kuwa ^2 x 2 - 5 x -3.

Zoezi$\PageIndex{23}$

Pata quotient:$\left(2 x^{2}-3 x-20\right) \div(x-4)$

Jibu

2x+5

Zoezi$\PageIndex{24}$

Pata quotient:$\left(3 x^{2}-16 x-12\right) \div(x-6)$

Jibu

3x+2

Wakati sisi kugawanywa 875 na 25, hatukuwa na salio. Lakini wakati mwingine mgawanyiko wa idadi haina kuondoka salio. Vile vile ni kweli wakati tunagawanya polynomials. Katika Zoezi$\PageIndex{25}$, tutakuwa na mgawanyiko kwamba majani salio. Tunaandika salio kama sehemu na mgawanyiko kama denominator.

Zoezi$\PageIndex{25}$

Pata quotient:$\left(x^{3}-x^{2}+x+4\right) \div(x+1)$

Jibu

Andika kama tatizo la mgawanyiko mrefu.
Kuwa na uhakika mgao ni katika hali ya kawaida.
Gawanya x ^3 na x. Weka jibu, x ^2, katika quotient juu ya muda x ^2. Kuzidisha x mara ^2 x + 1. Line up maneno kama chini ya mgao.
Ondoa x ^3 + x ^2 kutoka x ^3 - x ^2 kwa kubadilisha ishara na kuongeza. Kisha kuleta chini ya muda ujao.
Gawanya -1 x ^2 na x. Weka jibu, -1 x, katika quotient juu ya muda x. Kuzidisha -1 x mara x + 1. Line up maneno kama chini ya mgao.
Ondoa -2 x ^2 - 2 x kutoka -2 x ^2 + x kwa kubadilisha ishara na kuongeza. Kisha kuleta chini muda wa mwisho.
Gawanya 3 x na x. Weka jibu, 3, katika quotient juu ya muda wa mara kwa mara. Panua mara 3 x + 1. Line up maneno kama chini ya mgao.
Ondoa 3 x + 3 kutoka 3 x + 4 kwa kubadilisha ishara na kuongeza. Andika salio kama sehemu na mgawanyiko kama denominator.
Kuangalia,$(x+1)\left(x^{2}-2 x+3+\dfrac{1}{x+1}\right)$ kuzidisha Matokeo lazima$x^{3}-x^{2}+x+4$

Zoezi$\PageIndex{26}$

Pata quotient:$\left(x^{3}+5 x^{2}+8 x+6\right) \div(x+2)$

Jibu

$x^{2}+3 x+2+\dfrac{2}{x+2}$

Zoezi$\PageIndex{27}$

Pata quotient:$\left(2 x^{3}+8 x^{2}+x-8\right) \div(x+1)$

Jibu

$2 x^{2}+6 x-5-\dfrac{3}{x+1}$

Angalia nyuma kwenye gawio katika Mfano, Mfano, na Mfano. Masharti yaliandikwa katika utaratibu wa kushuka kwa digrii, na hapakuwa na digrii zilizopo. mgao katika Mfano itakuwa$x^{4}-x^{2}+5 x-2$. Inakosa$x^{3}$ muda. Sisi kuongeza katika$0x^{3}$ kama kishika.

Zoezi$\PageIndex{28}$

Pata quotient:$\left(x^{4}-x^{2}+5 x-2\right) \div(x+2)$

Jibu

Kumbuka kwamba hakuna$x^{3}$ mrefu katika mgao. Sisi kuongeza$0x^{3}$ kama kishika.

Andika kama tatizo la mgawanyiko mrefu. Kuwa na uhakika mgao ni katika hali ya kawaida na placeholders kwa masharti kukosa.
Gawanya x ^4 na x. Weka jibu, x ^3, katika quotient juu ya muda x ^3. Kuzidisha x mara ^3 x + 2. Weka masharti kama hayo. Ondoa na kisha kuleta chini ya muda ujao.
Gawanya -1 x ^3 na x. Weka jibu, -2 x ^2, katika quotient juu ya muda x ^2. Kuzidisha -1 x ^2 mara x + 1. Weka masharti kama hayo. Ondoa na kuleta chini ya muda ujao.
Gawanya 3 x ^2 na x. Weka jibu, 3 x, katika quotient juu ya muda x. Kuzidisha mara 3 x + 1. Weka masharti kama hayo. Ondoa na kuleta chini ya muda ujao.
Gawanya - x kwa x. Weka jibu, -1, katika quotient juu ya muda wa mara kwa mara. Kuzidisha -1 mara x + 1. Weka masharti kama hayo. Badilisha ishara, ongeza.
Kuangalia, kuzidisha$(x+2)\left(x^{3}-2 x^{2}+3 x-1\right)$
Matokeo yake yanapaswa kuwa$x^{4}-x^{2}+5 x-2$

Zoezi$\PageIndex{29}$

Pata quotient:$\left(x^{3}+3 x+14\right) \div(x+2)$

Jibu

$x^{2}-2 x+7$

Zoezi$\PageIndex{30}$

Pata quotient:$\left(x^{4}-3 x^{3}-1000\right) \div(x+5)$

Jibu

$x^{3}-8 x^{2}+40 x-200$

Katika Zoezi$\PageIndex{31}$, tutagawanya na$2a−3$. Kama sisi kugawanya tutakuwa na kuzingatia constants kama vile vigezo.

Zoezi$\PageIndex{31}$

Pata quotient:$\left(8 a^{3}+27\right) \div(2 a+3)$

Jibu

Wakati huu tutaonyesha mgawanyiko wote kwa hatua moja. Tunahitaji kuongeza placeholders mbili ili kugawanya.

Kuangalia, kuzidisha$(2 a+3)\left(4 a^{2}-6 a+9\right)$

Matokeo yake yanapaswa kuwa$8 a^{3}+27$

Zoezi$\PageIndex{32}$

Pata quotient:$\left(x^{3}-64\right) \div(x-4)$

Jibu

$x^{2}+4 x+16$

Zoezi$\PageIndex{33}$

Pata quotient:$\left(125 x^{3}-8\right) \div(5 x-2)$

Jibu

$25 x^{2}+10 x+4$

Kumbuka

Fikia rasilimali hizi za mtandaoni kwa maelekezo ya ziada na mazoezi na kugawa polynomials:

• Gawanya Polynomial na Monomial
• Gawanya Polynomial na Monomial 2
• Gawanya Polynomial na Binomial

Dhana muhimu

• Sehemu ya kuongeza
  • Ikiwa a, b, na c ni namba ambapo$c\neq 0$, basi
    $\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}$ na$\dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}$
    
• Idara ya Polynomial na Monomial
  • Ili kugawanya polynomial na monomial, kugawanya kila neno la polynomial na monomial.