Sura ya 5 Mazoezi Mapitio

Sura ya Mapitio ya mazoezi

Kuongeza na Ondoa Polynomials

Kuamua Shahada ya Polynomials

Katika mazoezi yafuatayo, tambua aina ya polynomial.

1. \(16x^2−40x−25\)

2. \(5m+9\)

Jibu

binomial

3. \(−15\)

4. \(y^2+6y^3+9y^4\)

Jibu

nyingine polynomial

Kuongeza na Ondoa Polynomials

Katika mazoezi yafuatayo, ongeza au uondoe polynomials.

5. \(4p+11p\)

6. \(−8y^3−5y^3\)

Jibu

\(−13y^3\)

7. \((4a^2+9a−11)+(6a^2−5a+10)\)

8. \((8m^2+12m−5)−(2m^2−7m−1)\)

Jibu

\(6m^2+19m−4\)

9. \((y^2−3y+12)+(5y^2−9)\)

10. \((5u^2+8u)−(4u−7)\)

Jibu

\(5u^2+4u+7\)

11. Kupata jumla ya\(8q^3−27\) na\(q^2+6q−2\).

12. Kupata tofauti ya\(x^2+6x+8\) na\(x^2−8x+15\).

Jibu

\(2x^2−2x+23\)

Katika mazoezi yafuatayo, kurahisisha.

13. \(17mn^2−(−9mn^2)+3mn^2\)

14. \(18a−7b−21a\)

Jibu

\(−7b−3a\)

15. \(2pq^2−5p−3q^2\)

16. \((6a^2+7)+(2a^2−5a−9)\)

Jibu

\(8a^2−5a−2\)

17. \((3p^2−4p−9)+(5p^2+14)\)

18. \((7m^2−2m−5)−(4m^2+m−8)\)

Jibu

\(−3m+3\)

19. \((7b^2−4b+3)−(8b^2−5b−7)\)

20. Ondoa\((8y^2−y+9)\) kutoka\( (11y^2−9y−5) \)

Jibu

\(3y^2−8y−14\)

21. Kupata tofauti ya\((z^2−4z−12)\) na\((3z^2+2z−11)\)

22. \((x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)\)

Jibu

\(x^3+2x^2y−4xy^2\)

23. \((x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)\)

Tathmini Kazi ya Polynomial kwa Thamani iliyotolewa ya Variable

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

24. Kwa kazi\(f(x)=7x^2−3x+5\) kupata:
a.\(f(5)\) b.\(f(−2)\) c.\(f(0)\)

Jibu

a. 165 b. 39 c 5

25. Kwa kazi\(g(x)=15−16x^2\), tafuta:
a.\(g(−1)\) b.\(g(0)\) c.\(g(2)\)

26. Miwani miwili imeshuka kwenye daraja 640 miguu juu ya mto. Kazi ya polynomial\(h(t)=−16t^2+640\) inatoa urefu wa glasi t sekunde baada ya kushuka. Kupata urefu wa glasi wakati\(t=6\).

Jibu

Urefu ni futi 64.

27. Mtengenezaji wa viatu vya hivi karibuni vya soka amegundua kwamba mapato yaliyopatikana kutokana na kuuza viatu kwa gharama ya\(p\) dola kila mmoja hutolewa na polynomial\(R(p)=−5p^2+360p\). Kupata mapato ya kupokea wakati\(p=110\) dola.

Ongeza na Ondoa Kazi za Polynomial

Katika mazoezi yafuatayo, tafuta.\((f + g)(x)\) b.\((f + g)(3)\) c.\((f − g)(x\) d.\((f − g)(−2)\)

28. \(f(x)=2x^2−4x−7\)na\(g(x)=2x^2−x+5\)

Jibu

a.\((f+g)(x)=4x^2−5x−2\)
b.\((f+g)(3)=19\)
c.\((f−g)(x)=−3x−12\)
d.\((f−g)(−2)=−6\)

29. \(f(x)=4x^3−3x^2+x−1\)na\(g(x)=8x^3−1\)

Mali ya Watazamaji na Nukuu za kisayansi

Kurahisisha Maneno Kutumia Mali kwa Watazamaji

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia mali kwa exponents.

30. \(p^3·p^{10}\)

Jibu

\(p^{13}\)

31. \(2·2^6\)

32. \(a·a^2·a^3\)

Jibu

\(a^6\)

33. \(x·x^8\)

34. \(y^a·y^b\)

Jibu

\(y^{a+b}\)

35. \(\dfrac{2^8}{2^2}\)

36. \(\dfrac{a^6}{a}\)

Jibu

\(a^5\)

37. \(\dfrac{n^3}{n^{12}}\)

38. \(\dfrac{1}{x^5}\)

Jibu

\(\dfrac{1}{x^4}\)

39. \(3^0\)

40. \(y^0\)

Jibu

\(1\)

41. \((14t)^0\)

42. \(12a^0−15b^0\)

Jibu

\(−3\)

Tumia Ufafanuzi wa Mtazamaji Mbaya

Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

43. \(6^{−2}\)

44. \((−10)^{−3}\)

Jibu

\(−\dfrac{1}{1000}\)

45. \(5·2^{−4}\)

46. \((8n)^{−1}\)

Jibu

\(\dfrac{1}{8n}\)

47. \(y^{−5}\)

48. \(10^{−3}\)

Jibu

\(\dfrac{1}{1000}\)

49. \(\dfrac{1}{a^{−4}}\)

50. \(\dfrac{1}{6^{−2}}\)

Jibu

\(36\)

51. \(−5^{−3}\)

52. \( \left(−\dfrac{1}{5}\right)^{−3}\)

Jibu

\(−\dfrac{1}{25}\)

53. \(−(12)^{−3}\)

54. \((−5)^{−3}\)

Jibu

\(−\dfrac{1}{125}\)

55. \(\left(\dfrac{5}{9}\right)^{−2}\)

56. \(\left(−\dfrac{3}{x}\right)^{−3}\)

Jibu

\(\dfrac{x^3}{27}\)

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Mali ya Bidhaa.

57. \((y^4)^3\)

58. \((3^2)^5\)

Jibu

\(3^{10}\)

59. \((a^{10})^y\)

60. \(x^{−3}·x^9\)

Jibu

\(x^5\)

61. \(r^{−5}·r^{−4}\)

62. \((uv^{−3})(u^{−4}v^{−2})\)

Jibu

\(\dfrac{1}{u^3v^5}\)

63. \((m^5)^{−1}\)

64. \(p^5·p^{−2}·p^{−4}\)

Jibu

\(\dfrac{1}{m^5}\)

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Power Mali.

65. \((k−2)^{−3}\)

66. \(\dfrac{q^4}{q^{20}}\)

Jibu

\(\dfrac{1}{q^{16}}\)

67. \(\dfrac{b^8}{b^{−2}}\)

68. \(\dfrac{n^{−3}}{n^{−5}}\)

Jibu

\(n^2\)

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Bidhaa kwa Power Mali.

69. \((−5ab)^3\)

70. \((−4pq)^0\)

Jibu

\(1\)

71. \((−6x^3)^{−2}\)

72. \((3y^{−4})^2\)

Jibu

\(\dfrac{9}{y^8}\)

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia Quotient kwa Power Mali.

73. \(\left(\dfrac{3}{5x}\right)^{−2}\)

74. \(\left(\dfrac{3xy^2}{z}\right)^4\)

Jibu

\(\dfrac{81x^4y^8}{z^4}\)

75. \((4p−3q^2)^2\)

Katika mazoezi yafuatayo, kurahisisha kila kujieleza kwa kutumia mali kadhaa.

76. \((x^2y)^2(3xy^5)^3\)

Jibu

\(27x^7y^{17}\)

77. \((−3a^{−2})^4(2a^4)^2(−6a^2)^3\)

78. \(\left(\dfrac{3xy^3}{4x^4y^{−2}}\right)^2\left(\dfrac{6xy^4}{8x^3y^{−2}}\right)^{−1}\)

Jibu

\(\dfrac{3y^4}{4x^4}\)

Katika mazoezi yafuatayo, weka kila nambari katika maelezo ya kisayansi.

79. \(2.568\)

80. \(5,300,000\)

Jibu

\(5.3×10^6\)

81. \(0.00814\)

Katika mazoezi yafuatayo, kubadilisha kila nambari kwa fomu ya decimal.

82. \(2.9×10^4\)

Jibu

\(29,000\)

83. \(3.75×10^{−1}\)

84. \(9.413×10^{−5}\)

Jibu

\(0.00009413\)

Katika mazoezi yafuatayo, kuzidisha au kugawanya kama ilivyoonyeshwa. Andika jibu lako kwa fomu ya decimal.

85. \((3×10^7)(2×10^{−4})\)

86. \((1.5×10^{−3})(4.8×10^{−1})\)

Jibu

\(0.00072\)

87. \(\dfrac{6×10^9}{2×10^{−1}}\)

88. \(\dfrac{9×10^{−3}}{1×10^{−6}}\)

Jibu

\(9,000\)

Kuzidisha Polynomials

Kuzidisha Monomials

Katika mazoezi yafuatayo, kuzidisha monomials.

89. \((−6p^4)(9p)\)

90. \(\left(\frac{1}{3}c^2\right)(30c^8)\)

Jibu

\(10c^{10}\)

91. \((8x^2y^5)(7xy^6)\)

92. \( \left(\frac{2}{3}m^3n^6\right)\left(\frac{1}{6}m^4n^4\right)\)

Jibu

\(\dfrac{m^7n^{10}}{9}\)

Kuzidisha Polynomial na Monomial

Katika mazoezi yafuatayo, ongeze.

93. \(7(10−x)\)

94. \(a^2(a^2−9a−36)\)

Jibu

\(a^4−9a^3−36a^2\)

95. \(−5y(125y^3−1)\)

96. \((4n−5)(2n^3)\)

Jibu

\(8n^4−10n^3\)

Kuzidisha Binomial na Binomial

Katika mazoezi yafuatayo, kuzidisha binomials kutumia:

a. Mali ya Kusambaza b. njia ya FOIL c. Njia ya Wima.

97. \((a+5)(a+2)\)

98. \((y−4)(y+12)\)

Jibu

\(y^2+8y−48\)

99. \((3x+1)(2x−7)\)

100. \((6p−11)(3p−10)\)

Jibu

\(18p^2−93p+110\)

Katika mazoezi yafuatayo, kuzidisha binomials. Tumia njia yoyote.

101. \((n+8)(n+1)\)

102. \((k+6)(k−9)\)

Jibu

\(k^2−3k−54\)

103. \((5u−3)(u+8)\)

104. \((2y−9)(5y−7)\)

Jibu

\(10y^2−59y+63\)

105. \((p+4)(p+7)\)

106. \((x−8)(x+9)\)

Jibu

\(x^2+x−72\)

107. \((3c+1)(9c−4)\)

108. \((10a−1)(3a−3)\)

Jibu

\(30a^2−33a+3\)

Kuzidisha Polynomial na Polynomial

Katika mazoezi yafuatayo, kuzidisha kwa kutumia a. Mali ya Usambazaji b. Njia ya Wima.

109. \((x+1)(x^2−3x−21)\)

110. \((5b−2)(3b^2+b−9)\)

Jibu

\(15b^3−b^2−47b+18\)

Katika mazoezi yafuatayo, ongeze. Tumia njia yoyote.

111. \((m+6)(m^2−7m−30)\)

112. \((4y−1)(6y^2−12y+5)\)

Jibu

\(24y^2−54y^2+32y−5\)

Kuzidisha Bidhaa Maalum

Katika mazoezi yafuatayo, mraba kila binomial kwa kutumia Pattern ya Mraba ya Binomial.

113. \((2x−y)^2\)

114. \((x+\dfrac{3}{4})^2\)

Jibu

\(x^2+\dfrac{3}{2}x+\dfrac{9}{16}\)

115. \((8p^3−3)^2\)

116. \((5p+7q)^2\)

Jibu

\(25p^2+70pq+49q^2\)

Katika mazoezi yafuatayo, kuzidisha kila jozi ya conjugates kwa kutumia Bidhaa ya Conjugates.

117. \((3y+5)(3y−5)\)

118. \((6x+y)(6x−y)\)

Jibu

\(36x^2−y^2\)

119. \((a+\dfrac{2}3b)(a−\dfrac{2}{3}b)\)

120. \((12x^3−7y^2)(12x^3+7y^2)\)

Jibu

\(144x^6−49y^4\)

121. \((13a^2−8b4)(13a^2+8b^4)\)

Gawanya Monomials

Kugawanya Monomials

Katika mazoezi yafuatayo, ugawanye monomials.

122. \(72p^{12}÷8p^3\)

Jibu

\(9p^9\)

123. \(−26a^8÷(2a^2)\)

124. \(\dfrac{45y^6}{−15y^{10}}\)

Jibu

\(−3y^4\)

125. \(\dfrac{−30x^8}{−36x^9}\)

126. \(\dfrac{28a^9b}{7a^4b^3}\)

Jibu

\(\dfrac{4a^5}{b^2}\)

127. \(\dfrac{11u^6v^3}{55u^2v^8}\)

128. \(\dfrac{(5m^9n^3)(8m^3n^2)}{(10mn^4)(m^2n^5)}\)

Jibu

\(\dfrac{4m^9}{n^4}\)

129. \(\dfrac{(42r^2s^4)(54rs^2)}{(6rs^3)(9s)}\)

Gawanya Polynomial na Monomial

Katika mazoezi yafuatayo, ugawanye kila polynomial na monomial

130. \((54y^4−24y^3)÷(−6y^2)\)

Jibu

\(−9y^2+4y\)

131. \(\dfrac{63x^3y^2−99x^2y^3−45x^4y^3}{9x^2y^2}\)

132. \(\dfrac{12x^2+4x−3}{−4x}\)

Jibu

\(−3x−1+\dfrac{3}{4x}\)

Gawanya Polynomials kutumia Long Idara

Katika mazoezi yafuatayo, kugawanya kila polynomial na binomial.

133. \((4x^2−21x−18)÷(x−6)\)

134. \((y^2+2y+18)÷(y+5)\)

Jibu

\(y−3+\dfrac{33}{q+6}\)

135. \((n^3−2n^2−6n+27)÷(n+3)\)

136. \((a^3−1)÷(a+1)\)

Jibu

\(a^2+a+1\)

Gawanya Polynomials kwa kutumia Division

Katika mazoezi yafuatayo, tumia Idara ya synthetic ili kupata quotient na salio.

137. \(x^3−3x^2−4x+12\)imegawanywa na\(x+2\)

138. \(2x^3−11x^2+11x+12\)imegawanywa na\(x−3\)

Jibu

\(2x^2−5x−4;\space0\)

139. \(x^4+x^2+6x−10\)imegawanywa na\(x+2\)

Gawanya Kazi za Polynomial

Katika mazoezi yafuatayo, ugawanye.

140. Kwa ajili ya kazi\(f(x)=x^2−15x+45\) na\(g(x)=x−9\), kupata.\(\left(\dfrac{f}{g}\right)(x)\)
b.\(\left(\dfrac{f}{g}\right)(−2)\)

Jibu

a.\(\left(\dfrac{f}{g}\right)(x)=x−6\)
b.\(\left(\dfrac{f}{g}\right)(−2)=−8\)

141. Kwa ajili ya kazi\(f(x)=x^3+x^2−7x+2\) na\(g(x)=x−2\), kupata.\(\left(\dfrac{f}{g}\right)(x)\)
b.\(\left(\dfrac{f}{g}\right)(3)\)

Tumia Theorem ya Salio na Sababu

Katika mazoezi yafuatayo, tumia Theorem ya Salio ili kupata salio.

142. \(f(x)=x^3−4x−9\)imegawanywa na\(x+2\)

Jibu

\(−9\)

143. \(f(x)=2x^3−6x−24\)kugawanywa na\(x−3\)

Katika mazoezi yafuatayo, tumia Theorem ya Factor kuamua ikiwa\(x−c\) ni sababu ya kazi ya polynomial.

144. Kuamua kama\(x−2\) ni sababu ya\(x^3−7x^2+7x−6\)

Jibu

hapana

145. Kuamua kama\(x−3\) ni sababu ya\(x^3−7x^2+11x+3\)

Sura ya Mazoezi mtihani

1. Kwa polynomial\(8y^4−3y^2+1\)

a Je, ni monomial, binomial, au trinomial? b. shahada yake ni nini?

Jibu

a. trinomial b. 4

2. \((5a^2+2a−12)(9a^2+8a−4)\)

3. \((10x^2−3x+5)−(4x^2−6)\)

Jibu

\(6x^2−3x+11\)

4. \(\left(−\dfrac{3}{4}\right)^3\)

5. \(x^{−3}x^4\)

Jibu

\(x\)

6. \(5^65^8\)

7. \((47a^{18}b^{23}c^5)^0\)

Jibu

\(1\)

8. \(4^{−1}\)

9. \((2y)^{−3}\)

Jibu

\(\dfrac{1}{8y^3}\)

10. \(p^{−3}·p^{−8}\)

11. \(\dfrac{x^4}{x^{−5}}\)

Jibu

\(x^9\)

12. \((3x^{−3})^2\)

13. \(\dfrac{24r^3s}{6r^2s^7}\)

Jibu

\(\dfrac{4r}{s^6}\)

14. \((x4y9x−3)2\)

15. \((8xy^3)(−6x^4y^6)\)

Jibu

\(−48x^5y^9\)

16. \(4u(u^2−9u+1)\)

17. \((m+3)(7m−2)\)

Jibu

\(21m^2−19m−6\)

18. \((n−8)(n^2−4n+11)\)

19. \((4x−3)^2\)

Jibu

\(16x^2−24x+9\)

20. \((5x+2y)(5x−2y)\)

21. \((15xy^3−35x^2y)÷5xy\)

Jibu

\(3y^2−7x \)

22. \((3x^3−10x^2+7x+10)÷(3x+2)\)

23. Matumizi Factor Theorem kuamua\(x+3\) kama sababu ya\(x^3+8x^2+21x+18\).

Jibu

ndiyo

24. a. Badilisha 112,000 kwa nukuu ya kisayansi.
b Badilisha kwenye fomu\(5.25×10^{−4}\) ya decimal.

Katika mazoezi yafuatayo, kurahisisha na kuandika jibu lako kwa fomu ya decimal.

25. \((2.4×10^8)(2×10^{−5})\)

Jibu

\(4.4×10^3\)

26. \(\dfrac{9×10^4}{3×10^{−1}}\)

27. Kwa kazi\(f(x)=6x^2−3x−9\) kupata:
a.\(f(3)\) b.\(f(−2)\) c.\(f(0)\)

Jibu

a.\(36\) b.\(21\) c.\(-9\)

28. Kwa\(f(x)=2x^2−3x−5\) na\(g(x)=3x^2−4x+1\),
kupata.\((f+g)(x)\) b.\((f+g)(1)\)
c.\((f−g)(x)\) d.\((f−g)(−2)\)

29. Kwa ajili ya kazi\(f(x)=3x^2−23x−36\) na\(g(x)=x−9\),
kupata.\(\left(\dfrac{f}{g}\right)(x)\) b.\(\left(\dfrac{f}{g}\right)(3)\)

Jibu

a.\(\left(\dfrac{f}{g}\right)(x)=3x+4\)
b.\(\left(\dfrac{f}{g}\right)(3)=13\)

30. Hiker matone majani kutoka daraja\(240\) miguu juu ya korongo. Kazi\(h(t)=−16t^2+240\) hutoa urefu wa\(t\) sekunde za majani baada ya kushuka. Kupata urefu wakati\(t=3\).