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10.E: Polynomials (Mazoezi)

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    10.1 - Ongeza na Ondoa Polynomials

    Kutambua Polynomials, Monomials, Binomials na Trinomials

    Katika mazoezi yafuatayo, onyesha kama kila moja ya polynomials zifuatazo ni monomial, binomial, trinomial, au polynomial nyingine.

    1. y 2 + 8y - 20
    2. -6a 4
    3. 9x 3 - 1
    4. n 3 - 3n 2 + 3n - 1

    Kuamua Shahada ya Polynomials

    Katika mazoezi yafuatayo, tambua kiwango cha kila polynomial.

    1. 16x 2 - 40x - 25
    2. 5m + 9
    3. -15
    4. na 2 + 6y 3 + 9y 4

    Kuongeza na Ondoa Monomials

    Katika mazoezi yafuatayo, ongeza au uondoe monomials.

    1. 4p + 11p
    2. -8y 3 - 5y 3
    3. Ongeza 4n 5, -n 5, -6n 5
    4. Ondoa 10x 2 kutoka 3x 2

    Kuongeza na Ondoa Polynomials

    Katika mazoezi yafuatayo, ongeza au uondoe polynomials.

    1. (4a 2 + 9a - 11) + (6a 2 - 5a + 10)
    2. (8m 2 + 12m - 5) - (2m 2 - 7m - 1)
    3. (y 2 - 3y + 12) + (5y 2 - 9)
    4. (5u 2 + 8u) - (4u - 7)
    5. Pata jumla ya 8q 3 - 27 na q 2 + 6q - 2
    6. Pata tofauti ya x 2 + 6x + 8 na x 2 - 8x + 15

    Tathmini Polynomial kwa Thamani iliyotolewa ya Variable

    Katika mazoezi yafuatayo, tathmini kila polynomial kwa thamani iliyotolewa.

    1. 200x -\(\dfrac{1}{5} x^{2}\) wakati x = 5
    2. 200x -\(\dfrac{1}{5} x^{2}\) wakati x = 0
    3. 200x -\(\dfrac{1}{5} x^{2}\) wakati x = 15
    4. 5 + 40x -\(\dfrac{1}{2} x^{2}\) wakati x = 10
    5. 5 + 40x -\(\dfrac{1}{2} x^{2}\) wakati x = -4
    6. 5 + 40x -\(\dfrac{1}{2} x^{2}\) wakati x = 0
    7. Miwani miwili imeshuka kwenye daraja 640 miguu juu ya mto. Polynomial -16t 2 + 640 inatoa urefu wa glasi t sekunde baada ya kushuka. Pata urefu wa glasi wakati t = 6.
    8. Ufanisi wa mafuta (kwa maili kwa kila lita) ya basi inayoenda kwa kasi ya maili x kwa saa hutolewa na polynomial\(− \dfrac{1}{160} x^{2} + \dfrac{1}{2} x\). Kupata ufanisi wa mafuta wakati x = 20 mph.

    10.2 - Tumia Mali ya kuzidisha ya Watazamaji

    Kurahisisha Maneno na Watazamaji

    Katika mazoezi yafuatayo, kurahisisha.

    1. 6 3
    2. \(\left(\dfrac{1}{2}\right)^{4}\)
    3. (-0.5) 2
    4. -3 2

    Kurahisisha Maneno Kutumia Mali ya Bidhaa ya Watazamaji

    Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

    1. p 3 • p 10
    2. 2 • 2 6
    3. a • a 2 • a 3
    4. x • x 8

    Kurahisisha Maneno Kutumia Mali ya Nguvu ya Wasanii

    Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

    1. (na 43)
    2. (r 32)
    3. (3 2)
    4. (a 10) y

    Kurahisisha Maneno Kutumia Bidhaa kwa Mali ya Nguvu

    Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

    1. (8n) 2
    2. (-5x 3)
    3. (2ab) 8
    4. (-10mnp) 4

    Kurahisisha Maneno kwa kutumia Mali kadhaa

    Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

    1. (3a 3)
    2. (4y) 2 (8y)
    3. (x 3) 5 (x 2) 3
    4. (5st 2) 3 (2s 3 hadi 4) 2

    Kuzidisha Monomials

    Katika mazoezi yafuatayo, kuzidisha monomials.

    1. (-6p 4) (9p)
    2. \(\left(\dfrac{1}{3} c^{2}\right)\)(30c 8)
    3. (8x 2 y 5) (7x 6)
    4. \(\left(\dfrac{2}{3} m^{3} n^{6}\right) \left(\dfrac{1}{6} m^{4} n^{4}\right)\)

    10.3 - Kuzidisha Polynomials

    Kuzidisha Polynomial na Monomial

    Katika mazoezi yafuatayo, ongeze.

    1. 7 (10 - x)
    2. a 2 (a 2 - 9a - 36)
    3. -5y (125y 3 - 1)
    4. (4n - 5) (2n 3)

    Kuzidisha Binomial na Binomial

    Katika mazoezi yafuatayo, kuzidisha binomials kwa kutumia mbinu mbalimbali.

    1. (a + 5) (a + 2)
    2. (y - 4) (y + 12)
    3. (3x + 1) (2x - 7)
    4. (6p - 11) (3p - 10)
    5. (n + 8) (n + 1)
    6. (k + 6) (k - 9)
    7. (5u - 3) (u + 8)
    8. (2y - 9) (5y - 7)
    9. (p + 4) (p + 7)
    10. (x - 8) (x + 9)
    11. (3c + 1) (9c - 4)
    12. (10a - 1) (3a - 3)

    Kuzidisha Trinomial na Binomial

    Katika mazoezi yafuatayo, kuzidisha kutumia njia yoyote.

    1. (x + 1) (x 2 - 3x - 21)
    2. (5b - 2) (3b 2 + b - 9)
    3. (m + 6) (m 2 - 7m - 30)
    4. (4y - 1) (6y 2 - 12y + 5)

    10.4 - Gawanya Monomials

    Kurahisisha Maneno Kutumia Mali ya Quotient ya Watazamaji

    Katika mazoezi yafuatayo, kurahisisha.

    1. \(\dfrac{2^{8}}{2^{2}}\)
    2. \(\dfrac{a^{6}}{a}\)
    3. \(\dfrac{n^{3}}{n^{12}}\)
    4. \(\dfrac{x}{x^{5}}\)

    Kurahisisha Maneno na Zero Exponents

    Katika mazoezi yafuatayo, kurahisisha.

    1. 3 0
    2. na 0
    3. (14t)
    4. 12a 0 - 15b 0

    Kurahisisha Maneno Kutumia Quotient kwa Mali ya Nguvu

    Katika mazoezi yafuatayo, kurahisisha.

    1. \(\left(\dfrac{3}{5}\right)^{2}\)
    2. \(\left(\dfrac{x}{2}\right)^{5}\)
    3. \(\left(\dfrac{5m}{n}\right)^{3}\)
    4. \(\left(\dfrac{s}{10t}\right)^{2}\)

    Kurahisisha Maneno kwa kutumia Mali kadhaa

    Katika mazoezi yafuatayo, kurahisisha.

    1. \(\dfrac{(a^{3})^{2}}{a^{4}}\)
    2. \(\dfrac{u^{3}}{u^{2} \cdot u^{4}}\)
    3. \(\left(\dfrac{x}{x^{9}}\right)^{5}\)
    4. \(\left(\dfrac{p^{4} \cdot p^{5}}{p^{3}}\right)^{2}\)
    5. \(\dfrac{(n^{5})^{3}}{(n^{2})^{8}}\)
    6. \(\left(\dfrac{5s^{2}}{4t}\right)^{3}\)

    Gawanya Monomials

    Katika mazoezi yafuatayo, ugawanye monomials.

    1. 72p 12 ÷ 8p 3
    2. -26a 8 ÷ (2a 2)
    3. \(\dfrac{45y^{6}}{−15y^{10}}\)
    4. \(\dfrac{−30x^{8}}{−36x^{9}}\)
    5. \(\dfrac{28a^{9} b}{7a^{4} b^{3}}\)
    6. \(\dfrac{11u^{6} v^{3}}{55u^{2} v^{8}}\)
    7. \(\dfrac{(5m^{9} n^{3})(8m^{3} n^{2})}{(10mn^{4})(m^{2} n^{5})}\)
    8. \(\dfrac{42r^{2} s^{4}}{6rs^{3}} − \dfrac{54rs^{2}}{9s}\)

    10.5 - Integer Exponents na Nukuu ya kisayansi

    Tumia Ufafanuzi wa Mtazamo Mbaya

    Katika mazoezi yafuatayo, kurahisisha.

    1. 6 -2
    2. (-10) —3
    3. 5 • 2 -4
    4. (8n) -1

    Kurahisisha Maneno na Exponents Integer

    Katika mazoezi yafuatayo, kurahisisha.

    1. x 1-3 • x 9
    2. r -5 •r -4
    3. (Uv -3) (u -4 v -2)
    4. (m 5) -1
    5. (k -2) —3
    6. \(\dfrac{q^{4}}{q^{20}}\)
    7. \(\dfrac{b^{8}}{b^{−2}}\)
    8. \(\dfrac{n^{−3}}{n^{−5}}\)

    Badilisha kutoka Nukuu ya Decimal hadi Nukuu ya kisayansi

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

    1. 5,300,000
    2. 0.00814
    3. Unene wa kipande cha karatasi ni karibu 0.097 millimeter.
    4. Kulingana na www.cleanair.com, biashara za Marekani hutumia tani 21,000,000 za karatasi kwa mwaka.

    Badilisha Nukuu ya kisayansi kwa Fomu ya Decima

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

    1. 2.9 × 10 4
    2. 1.5 × 10 8
    3. 3.75 × 10 -1
    4. 9.413 × 10 -5

    Kuzidisha na Gawanya Kutumia Notation ya

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

    1. (3 × 10 7) (2 × 10 -4)
    2. (1.5 × 10 -3) (4.8 × 10 -1)
    3. \(\dfrac{6 \times 10^{9}}{2 \times 10^{−1}}\)
    4. \(\dfrac{9 \times 10^{-3}}{1 \times 10^{−6}}\)

    10.6 - Utangulizi wa Factoring Polynomials

    Pata sababu kubwa ya kawaida ya maneno mawili au Zaidi

    Katika mazoezi yafuatayo, pata sababu kubwa zaidi ya kawaida.

    1. 5m, 45
    2. 8a, 72
    3. 12x 2, 20x 3, 36x 4
    4. 9y 4, 21y 5, 15y 6

    Sababu ya Sababu kuu ya kawaida kutoka kwa Polynomial

    Katika mazoezi yafuatayo, fikiria sababu kubwa zaidi kutoka kwa kila polynomial.

    1. 16u - 24
    2. 15r + 35
    3. 6p 2 + 6p
    4. 10c 2 - 10c
    5. -9a 5 - 9a 3
    6. -7x 8 - 28x 3
    7. 5y 2 - 55y + 45
    8. 2q 5 - 16q 3 + 30q 2

    MTIHANI WA MAZOEZI

    1. Kwa 8y polynomial 4 - 3y 2 + 1
      1. Je, ni monomial, binomial, au trinomial?
      2. Shahada yake ni nini?

    Katika mazoezi yafuatayo, kurahisisha kila kujieleza.

    1. (5a 2 + 2a - 12) + (9a 2 + 8a - 4)
    2. (10x 2 - 3x + 5) - (4x 2 - 6)
    3. \(\left(− \dfrac{3}{4}\right)^{3}\)
    4. n • n 4
    5. (10p 3 q 5) 2
    6. (8x 3) (-6x 4 y 6)
    7. 4u (u 2 - 9u + 1)
    8. (s + 8) (s + 9)
    9. (m + 3) (7m - 2)
    10. (11a - 6) (5a - 1)
    11. (n - 8) (n 2 - 4n + 11)
    12. (4a + 9b) (6a - 5b)
    13. \(\dfrac{5^{6}}{5^{8}}\)
    14. \(\left(\dfrac{x^{3} \cdot x^{9}}{x^{5}}\right)^{2}\)
    15. (47a 18 b 23 c 5)
    16. \(\dfrac{24r^{3}s}{6r^{2} s^{7}}\)
    17. \(\dfrac{8y^{2} − 16y + 20}{4y}\)
    18. (15x 3 - 35x 2 y) ÷ 5x
    19. 4 -1
    20. (2y) —3
    21. p —3 • p -8
    22. \(\dfrac{x^{4}}{x^{−5}}\)
    23. (2.4 × 10 8) (2 × 10 -5)

    Katika mazoezi yafuatayo, fikiria sababu kubwa zaidi kutoka kwa kila polynomial.

    1. 80a 3 + 120a 2 + 40a
    2. -6x 2 - 30x
    3. Badilisha 5.25 × 10 -4 kwa fomu ya decimal.

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

    1. \(\dfrac{9 \times 10^{4}}{3 \times 10^{−1}}\)
    2. Hiker matone majani kutoka daraja 240 futi juu ya korongo. Polynomial -16t 2 + 240 inatoa urefu wa majani t sekunde baada ya kushuka. Kupata urefu wakati t = 3.
    3. Kulingana na www.cleanair.org, kiasi cha takataka kilichozalishwa nchini Marekani kwa mwaka mmoja kina wastani wa paundi 112,000 za takataka kwa kila mtu. Andika nambari hii kwa notation ya kisayansi.

    Wachangiaji na Majina