Sura ya 6 Mazoezi Mapitio

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Sura ya Mapitio ya mazoezi

Kipengele cha kawaida cha kawaida na Kikundi

Pata sababu kubwa ya kawaida ya maneno mawili au Zaidi

Katika mazoezi yafuatayo, pata sababu kubwa zaidi ya kawaida.

\(12a^2b^3,\space 15ab^2\)

Jibu: \(3ab^2\)

\(12m^2n^3,42m^5n^3\)

\(15y^3,\space 21y^2,\space 30y\)

Jibu: \(3y\)

\(45x^3y^2,\space 15x^4y,\space 10x^5y^3\)

Sababu ya Sababu kuu ya kawaida kutoka kwa Polynomial

Katika mazoezi yafuatayo, factor sababu kubwa ya kawaida kutoka kila polynomial.

\(35y+84\)

Jibu: \(7(5y+12)\)

\(6y^2+12y−6\)

\(18x^3−15x\)

Jibu: \(3x(6x^2−5)\)

\(15m^4+6m^2n\)

\(4x^3−12x^2+16x\)

Jibu: \(4x(x^2−3x+4)\)

\(−3x+24\)

\(−3x^3+27x^2−12x\)

Jibu: \(−3x(x^2−9x+4)\)

\(3x(x−1)+5(x−1)\)

Kipengele kwa Kundi

Katika mazoezi yafuatayo, sababu kwa kikundi.

\(ax−ay+bx−by\)

Jibu: \((a+b)(x−y)\)

\(x^2y−xy^2+2x−2y\)

\(x^2+7x−3x−21\)

Jibu: \((x−3)(x+7)\)

\(4x^2−16x+3x−12\)

\(m^3+m^2+m+1\)

Jibu: \((m^2+1)(m+1)\)

\(5x−5y−y+x\)

Factor Trinomials

Sababu Trinomials ya Fomu\(x^2+bx+c\)

Katika mazoezi yafuatayo, fanya kila trinomial ya fomu\(x^2+bx+c\).

\(a^2+14a+33\)

Jibu: \((a+3)(a+11)\)

\(k^2−16k+60\)

\(m^2+3m−54\)

Jibu: \((m+9)(m−6)\)

\(x^2−3x−10\)

Katika mifano ifuatayo, factor kila trinomial ya fomu\(x^2+bxy+cy^2\).

\(x^2+12xy+35y^2\)

Jibu: \((x+5y)(x+7y)\)

\(r^2+3rs−28s^2\)

\(a^2+4ab−21b^2\)

Jibu: \((a+7b)(a−3b)\)

\(p^2−5pq−36q^2\)

\(m^2−5mn+30n^2\)

Jibu: kuu

Sababu Trinomials ya Fomu ax2+bx+cax2+bx+c Kutumia Jaribio na Hitilafu

Katika mazoezi yafuatayo, factor kabisa kutumia jaribio na kosa.

\(x^3+5x^2−24x\)

\(3y^3−21y^2+30y\)

Jibu: \(3y(y−5)(y−2)\)

\(5x^4+10x^3−75x^2\)

\(5y^2+14y+9\)

Jibu: \((5y+9)(y+1)\)

\(8x^2+25x+3\)

\(10y^2−53y−11\)

Jibu: \((5y+1)(2y−11)\)

\(6p^2−19pq+10q^2\)

\(−81a^2+153a+18\)

Jibu: \(−9(9a−1)(a+2)\)

Sababu Trinomials ya Fomu ax2+bx+cax2+bx+c kwa kutumia njia ya 'ac'

Katika mazoezi yafuatayo, sababu.

\(2x^2+9x+4\)

\(18a^2−9a+1\)

Jibu: \((3a−1)(6a−1)\)

\(15p^2+2p−8\)

\(15x^2+6x−2\)

Jibu: \((3x−1)(5x+2)\)

\(8a^2+32a+24\)

\(3x^2+3x−36\)

Jibu: \(3(x+4)(x−3)\)

\(48y^2+12y−36\)

\(18a^2−57a−21\)

Jibu: \(3(2a−7)(3a+1)\)

\(3n^4−12n^3−96n^2\)

Sababu ya kutumia badala

Katika mazoezi yafuatayo, sababu ya kutumia badala.

\(x^4−13x^2−30\)

Jibu: \((x^2−15)(x^2+2)\)

\((x−3)^2−5(x−3)−36\)

Factor Maalum Bidhaa

Factor Perfect Square trinomials

Katika mazoezi yafuatayo, factor kabisa kutumia kamili mraba trinomials mfano.

\(25x^2+30x+9\)

Jibu: \((5x+3)^2\)

\(36a^2−84ab+49b^2\)

\(40x^2+360x+810\)

Jibu: \(10(2x+9)^2\)

\(5k^3−70k^2+245k\)

\(75u^4−30u^3v+3u^2v^2\)

Jibu: \(3u^2(5u−v)^2\)

Sababu tofauti ya Viwanja

Katika mazoezi yafuatayo, sababu kabisa kutumia tofauti ya muundo wa mraba, ikiwa inawezekana.

\(81r^2−25\)

\(169m^2−n^2\)

Jibu: \((13m+n)(13m−n)\)

\(25p^2−1\)

\(9−121y^2\)

Jibu: \((3+11y)(3−11y)\)

\(20x^2−125\)

\(169n^3−n\)

Jibu: \(n(13n+1)(13n−1)\)

\(6p^2q^2−54p^2\)

\(24p^2+54\)

Jibu: \(6(4p^2+9)\)

\(49x^2−81y^2\)

\(16z^4−1\)

Jibu: \((2z−1)(2z+1)(4z^2+1)\)

\(48m^4n^2−243n^2\)

\(a^2+6a+9−9b^2\)

Jibu: \((a+3−3b)(a+3+3b)\)

\(x^2−16x+64−y^2\)

Kiasi cha Kiasi na Tofauti za Cubes

Katika mazoezi yafuatayo, factor kabisa kutumia kiasi na tofauti ya muundo wa cubes, ikiwa inawezekana.

\(a^3−125\)

Jibu: \((a−5)(a^2+5a+25)\)

\(b^3−216\)

\(2m^3+54\)

Jibu: \(2(m+3)(m^2−3m+9)\)

\(81m^3+3\)

General Mkakati wa Factoring Polynomials

Kutambua na Tumia Njia sahihi ya Kufanya Kipolynomial Kikamilifu

Katika mazoezi yafuatayo, factor kabisa.

\(24x^3+44x^2\)

Jibu: \(4x^2(6x+11)\)

\(24a^4−9a^3\)

\(16n^2−56mn+49m^2\)

Jibu: \((4n−7m)^2\)

\(6a^2−25a−9\)

\(5u^4−45u^2\)

Jibu: \(5u^2(u+3)(u−3)\)

\(n^4−81\)

\(64j^2+225\)

Jibu: mkuu

\(5x^2+5x−60\)

\(b^3−64\)

Jibu: \((b−4)(b^2+4b+16)\)

\(m^3+125\)

\(2b^2−2bc+5cb−5c^2\)

Jibu: \((2b+5c)(b−c)\)

\(48x^5y^2−243xy^2\)

\(5q^2−15q−90\)

Jibu: \(5(q+3)(q−6) \)

\(4u^5v+4u^2v^3\)

\(10m^4−6250\)

Jibu: \(10(m−5)(m+5)(m^2+25)\)

\(60x^2y−75xy+30y\)

\(16x^2−24xy+9y^2−64\)

Jibu: \((4x−3y+8)(4x−3y−8)\)

Ulinganifu wa Polynomial

Tumia mali ya Bidhaa ya Zero

Katika mazoezi yafuatayo, tatua.

\((a−3)(a+7)=0\)

\((5b+1)(6b+1)=0\)

Jibu: \(b=−\frac{1}{5},\space b=−\frac{1}{6}\)

\(6m(12m−5)=0\)

\((2x−1)^2=0\)

Jibu: \(x=\frac{1}{2}\)

\(3m(2m−5)(m+6)=0\)

Tatua Ulinganisho wa Quadratic kwa kuzingatia

Katika mazoezi yafuatayo, tatua.

\(x^2+9x+20=0\)

Jibu: \(x=−4,\space x=−5\)

\(y^2−y−72=0\)

\(2p^2−11p=40\)

Jibu: \(p=−\frac{5}{2},p=8\)

\(q^3+3q^2+2q=0\)

\(144m^2−25=0\)

Jibu: \(m=\frac{5}{12},\space m=−\frac{5}{12}\)

\(4n^2=36\)

\((x+6)(x−3)=−8\)

Jibu: \(x=2,\space x=−5\)

\((3x−2)(x+4)=12\)

\(16p^3=24p^2+9p\)

Jibu: \(p=0,\space p=\frac{3}{4}\)

\(2y^3+2y^2=12y\)

Tatua Ulinganisho na Kazi za Polynomial

Katika mazoezi yafuatayo, tatua.

Kwa ajili ya kazi\(f(x)=x^2+11x+20\),, ⓐ kupata wakati\(f(x)=−8\) ⓑ Matumizi habari hii kupata pointi mbili kwamba uongo juu ya graph ya kazi.

Jibu: ⓐ\(x=−7\) au\\(x=−4\)
ⓑ\((−7,−8)\)\((−4,−8)\)

Kwa ajili ya kazi\(f(x)=9x^2−18x+5\),, ⓐ kupata wakati\(f(x)=−3\) ⓑ Matumizi habari hii kupata pointi mbili kwamba uongo juu ya graph ya kazi.

\(f(x)=64x^2−49\)

\(f(x)=6x^2−13x−5\)

Kutatua Maombi yanayotokana na equations Quadratic

Katika mazoezi yafuatayo, tatua.

Bidhaa ya namba mbili za mfululizo ni 399. Kupata idadi.

Jibu: Idadi ni\(−21\) na\(−19\) au 19 na 21.

Eneo la patio mstatili umbo 432 futi za mraba. Urefu wa patio ni miguu 6 zaidi ya upana wake. Pata urefu na upana.

Ngazi hutegemea ukuta wa jengo. Urefu wa ngazi ni urefu wa miguu 9 kuliko umbali wa chini ya ngazi kutoka jengo hilo. Umbali wa juu ya ngazi hufikia upande wa jengo ni urefu wa miguu 7 kuliko umbali wa chini ya ngazi kutoka jengo hilo. Pata urefu wa pande zote tatu za pembetatu iliyoundwa na ngazi inayotegemea jengo hilo.

Jibu: Urefu ni 8, 15, na 17 ft.

Shruti ni kwenda kutupa mpira kutoka juu ya mwamba. Wakati yeye throws mpira kutoka 80 miguu juu ya ardhi, kazi\(h(t)=−16t^2+64t+80\) mifano urefu, h, ya mpira juu ya ardhi kama kazi ya muda, t. Find: ⓐ zeros ya kazi hii ambayo inatuambia wakati mpira hit ardhi. ⓑ wakati (s) mpira itakuwa 80 miguu juu ya ardhi. ⓒ urefu mpira itakuwa katika\(t=2\) sekunde ambayo ni wakati mpira itakuwa katika hatua yake ya juu.

Sura ya Mazoezi mtihani

Katika mazoezi yafuatayo, factor kabisa.

\(80a^2+120a^3\)

Jibu: \(40a^2(2+3a)\)

\(5m(m−1)+3(m−1)\)

\(x^2+13x+36\)

Jibu: \((x+7)(x+6)\)

\(p^2+pq−12q^2\)

\(xy−8y+7x−56\)

Jibu: \((x−8)(y+7)\)

\(40r^2+810\)

\(9s^2−12s+4\)

Jibu: \((3s−2)^2\)

\(6x^2−11x−10\)

\(3x^2−75y^2\)

Jibu: \(3(x+5y)(x−5y)\)

\(6u^2+3u−18\)

\(x^3+125\)

Jibu: \((x+5)(x^2−5x+25)\)

\(32x^5y^2−162xy^2\)

\(6x^4−19x^2+15\)

Jibu: \((3x^2−5)(2x^2−3)\)

\(3x^3−36x^2+108x\)

Katika mazoezi yafuatayo, tatua

\(5a^2+26a=24\)

Jibu: \(a=\frac{4}{5},\space a=−6\)

Bidhaa ya integers mbili mfululizo ni 156. Pata integers.

Eneo la kitanda cha mstatili ni inchi za mraba 168. Urefu wake ni urefu wa inchi mbili kuliko upana. Pata urefu na upana wa mahali pa mahali.

Jibu: Upana ni inchi 12 na urefu ni inchi 14.

Jing ni kwenda kutupa mpira kutoka balcony ya condo yake. Wakati yeye throws mpira kutoka 80 miguu juu ya ardhi, kazi\(h(t)=−16t^2+64t+80\) mifano urefu, h, ya mpira juu ya ardhi kama kazi ya muda, t. Find: ⓐ zeros ya kazi hii ambayo inatuambia wakati mpira hit ardhi. ⓑ wakati (s) mpira itakuwa 128 miguu juu ya ardhi. ⓒ urefu mpira itakuwa\(t=4\) sekunde.

Kwa ajili ya kazi\(f(x)=x^2−7x+5\),, ⓐ kupata wakati\(f(x)=−7\) ⓑ Matumizi habari hii kupata pointi mbili kwamba uongo juu ya graph ya kazi.

Jibu: ⓐ\(x=3\) au\(x=4\) ⓑ\((3,−7)\)\((4,−7)\)

faharasa

shahada ya equation ya polynomial: Kiwango cha equation ya polynomial ni kiwango cha polynomial.

equation ya polynomial: Equation polynomial ni equation ambayo ina kujieleza polynomial.

quadratic equation: Ulinganisho wa polynomial wa shahada mbili huitwa equations quadratic.

sifuri ya kazi: Thamani ya xx ambapo kazi ni 0, inaitwa sifuri ya kazi.

Zero Bidhaa Mali: Mali ya Bidhaa ya Zero inasema kwamba ikiwa bidhaa ya kiasi mbili ni sifuri, basi angalau moja ya kiasi ni sifuri.