5.2E: Mazoezi

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Mazoezi hufanya kamili

Kuamua Aina ya Polynomials

Katika mazoezi yafuatayo, onyesha kama polynomial ni monomial, binomial, trinomial, au polynomial nyingine. Pia kutoa shahada ya kila polynomial.

1. ⓐ$47x^5−17x^2y^3+y^2$
ⓑ$5c^3+11c^2−c−8$
ⓒ$59ab+13b$
ⓓ$4$
ⓔ$4pq+17$

Jibu: ⓐ trinomial, shahada 5
ⓑ nyingine polynomial, shahada 3
ⓒ binomial, shahada 2
ⓓ monomial, shahada 0
ⓔ binomial, shahada 2

2. ⓐ$x^2−y^2$
ⓑ$−13c^4$
ⓒ$a^2+2ab−7b^2$
ⓓ$4x^2y^2−3xy+8$
ⓔ$19$

3. ⓐ$8y−5x$
ⓑ$y^2−5yz−6z^2$
ⓒ$y^3−8y^2+2y−16$
ⓓ$81ab^4−24a^2b^2+3b$
ⓔ$−18$

Jibu: ⓐ binomial, shahada 1
ⓑ trinomial, shahada 2
ⓒ polynomial nyingine, shahada 3
ⓓ trinomial, shahada 5
ⓔ monomial, shahada 0

4. ⓐ$11y^2$
ⓑ$−73$
ⓒ$6x^2−3xy+4x−2y+y^2$
ⓓ$4y^2+17z^2$
ⓔ$5c^3+11c^2−c−8$

5. ⓐ$5a^2+12ab−7b^2$
ⓑ$18xy^2z$
ⓒ$5x+2$
ⓓ$y^3−8y^2+2y−16$
ⓔ$−24$

Jibu: ⓐ trinomial, shahada 2
ⓑ monomial, shahada 4
ⓒ binomial, shahada 1
ⓓ nyingine polynomial, shahada 3
ⓔ monomial, shahada 0

6. ⓐ$9y^3−10y^2+2y−6$
ⓑ$−12p^3q$
ⓒ$a^2+9ab+18b^2$
ⓓ$20x^2y^2−10a^2b^2+30$
ⓔ$17$

7. ⓐ$14s−29t$
ⓑ$z^2−5z−6$
ⓒ$y^3−8y^2z+2yz^2−16z^3$
ⓓ$23ab^2−14$
ⓔ$−3$

Jibu: ⓐ binomial, shahada 1
ⓑ trinomial, shahada 2
ⓒ polynomial nyingine, shahada 3
ⓓ binomial, shahada 3
ⓔ monomial, shahada 0

8. ⓐ$15xy$
ⓑ$15$
ⓒ$6x^2−3xy+4x−2y+y^2$
ⓓ$10p−9q$
ⓔ$m^4+4m^3+6m^2+4m+1$

Kuongeza na Ondoa Polynomials

Katika mazoezi yafuatayo, ongeza au uondoe monomials.

9. ⓐ$7x^2+5x^2$
ⓑ$4a−9a$

Jibu: ⓐ$12x^2$ ⓑ$−5a$

10. ⓐ$4y^3+6y^3$
ⓑ$−y−5y$

11. ⓐ$−12w+18w$
ⓑ$7x^2y−(−12x^2y)$

Jibu: ⓐ$6w$
ⓑ$19x^2y$

12. ⓐ$−3m+9m$
ⓑ$15yz^2−(−8yz^2)$

13. $7x^2+5x^2+4a−9a$

Jibu: $12x^2−5a$

14. $4y^3+6y^3−y−5y$

15. $−12w+18w+7x^2y−(−12x^2y)$

Jibu: $6w+19x^2y$

16. $−3m+9m+15yz^2−(−8yz^2)$

17. ⓐ$−5b−17b$
ⓑ$3xy−(−8xy)+5xy$

Jibu: ⓐ$−22b$
ⓑ$16xy$

18. ⓐ$−10x−35x$
ⓑ$17mn^2−(−9mn^2)+3mn^2$

19. ⓐ$12a+5b−22a$
ⓑ$pq^2−4p−3q^2$

Jibu: ⓐ$−10a+5b$
ⓑ$pq^2−4p−3q^2$

20. ⓐ$14x−3y−13x$
ⓑ$a^2b−4a−5ab^2$

21. ⓐ$2a^2+b^2−6a^2$
ⓑ$x^2y−3x+7xy^2$

Jibu: ⓐ$−4a^2+b^2$
ⓑ$x^2y−3x+7xy^2$

22. ⓐ$5u^2+4v^2−6u^2$
ⓑ$12a+8b$

23. ⓐ$xy^2−5x−5y^2$
ⓑ$19y+5z$

Jibu: ⓐ$xy^2−5x−5y^2$
ⓑ$19y+5z$

24. $12a+5b−22a+pq^2−4p−3q^2$

25. $14x−3y−13x+a^2b−4a−5ab^2$

Jibu: $x−3y+a^2b−4a−5ab^2$

26. $2a^2+b^2−6a^2+x^2y−3x+7xy^2$

27. $5u^2+4v^2−6u^2+12a+8b$

Jibu: $−u^2+4v^2+12a+8b$

28. $xy^2−5x−5y^2+19y+5z$

29. Ongeza:$4a,−3b,−8a$

Jibu: $−4a−3b$

30. Ongeza:$4x,3y,−3x$

31. Ondoa$5x^6$ kutoka$−12x^6$

Jibu: $−7x^6$

32. Ondoa$2p^4$ kutoka$−7p^4$

Katika mazoezi yafuatayo, ongeza polynomials.

33. $(5y^2+12y+4)+(6y^2−8y+7)$

Jibu: $11y^2+4y+11$

34. $(4y^2+10y+3)+(8y^2−6y+5)$

35. $(x^2+6x+8)+(−4x^2+11x−9)$

Jibu: $−3x^2+17x−1$

36. $(y^2+9y+4)+(−2y^2−5y−1)$

37. $(8x^2−5x+2)+(3x^2+3)$

Jibu: $11x^2−5x+5$

38. $(7x^2−9x+2)+(6x^2−4)$

39. $(5a^2+8)+(a^2−4a−9)$

Jibu: $6a^2−4a−1$

40. $(p^2−6p−18)+(2p^2+11)$

Katika mazoezi yafuatayo, toa polynomials.

41. $(4m^2−6m−3)−(2m^2+m−7)$

Jibu: $2m^2−7m+4$

42. $(3b^2−4b+1)−(5b^2−b−2)$

43. $(a^2+8a+5)−(a^2−3a+2)$

Jibu: $11a+3$

44. $(b^2−7b+5)−(b^2−2b+9)$

45. $(12s^2−15s)−(s−9)$

Jibu: $12s^2−14s+9$

46. $(10r^2−20r)−(r−8)$

Katika mazoezi yafuatayo, toa polynomials.

47. Ondoa$(9x^2+2)$ kutoka$(12x^2−x+6)$

Jibu: $3x^2−x+4$

48. Ondoa$(5y^2−y+12)$ kutoka$(10y^2−8y−20)$

49. Ondoa$(7w^2−4w+2)$ kutoka$(8w^2−w+6)$

Jibu: $w^2+3w+4$

50. Ondoa$(5x^2−x+12)$ kutoka$(9x^2−6x−20)$

Katika mazoezi yafuatayo, tafuta tofauti ya polynomials.

51. Kupata tofauti ya$(w^2+w−42)$ na$(w^2−10w+24)$

Jibu: $11w−64$

52. Kupata tofauti ya$(z^2−3z−18)$ na$(z^2+5z−20)$

Katika mazoezi yafuatayo, ongeza polynomials.

53. $(7x^2−2xy+6y^2)+(3x^2−5xy)$

Jibu: $10x^2−7xy+6y^2$

54. $(−5x^2−4xy−3y^2)+(2x^2−7xy)$

55. $(7m^2+mn−8n^2)+(3m^2+2mn)$

Jibu: $10m^2+3mn−8n^2$

56. $(2r^2−3rs−2s^2)+(5r^2−3rs)$

Katika mazoezi yafuatayo, ongeza au uondoe polynomials.

57. $(a^2−b^2)−(a^2+3ab−4b^2)$

Jibu: $−3ab+3b^2$

58. $(m^2+2n^2)−(m^2−8mn−n^2)$

59. $(p^3−3p^2q)+(2pq^2+4q^3)−(3p^2q+pq^2)$

Jibu: $p^3−6p^2q+pq^2+4q^3$

60. $(a^3−2a^2b)+(ab^2+b^3)−(3a^2b+4ab^2)$

61. $(x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)$

Jibu: $x^3+2x^2y−5xy^2+y^3$

62. $(x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)$

Tathmini Kazi ya Polynomial kwa Thamani iliyotolewa

Katika mazoezi yafuatayo, tafuta maadili ya kazi kwa kila kazi ya polynomial.

Katika mazoezi yafuatayo, pata urefu wa kila kazi ya polynomial.

67. Mchoraji matone brashi kutoka$75$ miguu jukwaa juu. Kazi ya polynomial$h(t)=−16t^2+75$ inatoa urefu wa$t$ sekunde za brashi baada ya kushuka. Pata urefu baada ya$t=2$ sekunde.

Jibu: Urefu ni futi 11.

68. msichana matone mpira mbali cliff ndani ya bahari. Polynomial$h(t)=−16t^2+200$ inatoa urefu wa$t$ sekunde za mpira baada ya kushuka. Pata urefu baada ya$t=3$ sekunde.

69. Mtengenezaji wa wasemaji wa sauti ya stereo amegundua kwamba mapato yaliyopatikana kutokana na kuuza wasemaji kwa gharama ya$p$ dola kila mmoja hutolewa na kazi ya polynomial$R(p)=−4p^2+420p$. Kupata mapato ya kupokea wakati$p=60$ dola.

Jibu: Mapato ni $10,800.

70. Mtengenezaji wa viatu vya hivi karibuni vya mpira wa kikapu amegundua kwamba mapato yaliyopatikana kutokana na kuuza viatu kwa gharama ya$p$ dola kila mmoja hutolewa na polynomial$R(p)=−4p^2+420p$. Kupata mapato ya kupokea wakati$p=90$ dola.

71. Polynomial$C(x)=6x^2+90x$ inatoa gharama, kwa dola, ya kuzalisha chombo cha mstatili ambao juu na chini ni mraba na miguu ya upande na$x$ miguu ya urefu$6$. Pata gharama ya kuzalisha sanduku na$x=4$ miguu.

Jibu: Gharama ni $456.

72. Polynomial$C(x)=6x^2+90x$ inatoa gharama, kwa dola, ya kuzalisha chombo cha mstatili ambao juu na chini ni mraba na miguu ya upande na$x$ miguu ya urefu$4$. Pata gharama ya kuzalisha sanduku na$x=6$ miguu.

Ongeza na Ondoa Kazi za Polynomial

73. $f(x)=2x^2−4x+1$na$g(x)=5x^2+8x+3$

74. $f(x)=4x^2−7x+3$na$g(x)=4x^2+2x−1$

75. $f(x)=3x^3−x^2−2x+3$na$g(x)=3x^3−7x$

76. $f(x)=5x^3−x^2+3x+4$na$g(x)=8x^3−1$

Mazoezi ya kuandika

77. Kutumia maneno yako mwenyewe, kuelezea tofauti kati ya monomial, binomial, na trinomial.

Jibu: Majibu yatatofautiana.

78. Kutumia maneno yako mwenyewe, kuelezea tofauti kati ya polynomial na maneno tano na polynomial na shahada ya$5$.

79. Ariana anadhani jumla$6y^2+5y^4$ ni$11y^6$. Ni nini kibaya na hoja yake?

Jibu: Majibu yatatofautiana.

80. Je, kila trinomial ni polynomial ya shahada ya pili? Ikiwa sio, fanya mfano.

Self Check

ⓐ Baada ya kukamilisha mazoezi, tumia orodha hii ili kutathmini ujuzi wako wa malengo ya sehemu hii.

$Takwimu inaonyesha meza yenye safu sita na nguzo nne. Mstari wa kwanza ni mstari wa kichwa na huandika kila safu. Kichwa cha kwanza cha safu ni “Naweza...”, pili ni “kwa ujasiri”, ya tatu ni “kwa msaada fulani”, “hakuna minus siipate!”. Chini ya safu ya kwanza ni maneno “kutambua polynomials, monomials, binomials, na trinomials”, “kuamua kiwango cha polynomials”, “kuongeza na kuondoa monomials”, “kuongeza na kuondoa polynomials”, na “tathmini polynomial kwa thamani fulani”. Chini ya nguzo ya pili, ya tatu, ya nne ni nafasi tupu ambapo mwanafunzi anaweza kuangalia kiwango gani cha ustadi waliyopata.$

ⓑ Kama wengi wa hundi yako walikuwa:

... kwa ujasiri. Hongera! Umefanikiwa malengo katika sehemu hii. Fikiria ujuzi wa kujifunza uliyotumia ili uweze kuendelea kuitumia. Ulifanya nini ili uwe na ujasiri wa uwezo wako wa kufanya mambo haya? Kuwa maalum.

... kwa msaada fulani. Hii lazima kushughulikiwa haraka kwa sababu mada huna bwana kuwa mashimo katika barabara yako ya mafanikio. Katika hesabu kila mada hujenga juu ya kazi ya awali. Ni muhimu kuhakikisha kuwa na msingi imara kabla ya kuendelea. Nani unaweza kuomba msaada? Washiriki wenzako na mwalimu ni rasilimali nzuri. Je, kuna mahali kwenye chuo ambapo waalimu hisabati zinapatikana? Je, ujuzi wako wa kujifunza unaweza kuboreshwa?

... hapana - Siipati! Hii ni ishara ya onyo na haipaswi kupuuza. Unapaswa kupata msaada mara moja au utazidiwa haraka. Angalia mwalimu wako haraka iwezekanavyo kujadili hali yako. Pamoja unaweza kuja na mpango wa kupata msaada unayohitaji.