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Expand the following: (m-2n)5, (3x+y)4, (2y-3x)3

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3 parts
a

m5−10m4n+40m3n2−80m2n3+80mn4−32n5m^5 - 10m^4n + 40m^3n^2 - 80m^2n^3 + 80mn^4 - 32n^5

b

81x4+108x3y+54x2y2+12xy3+y481x^4 + 108x^3y + 54x^2y^2 + 12xy^3 + y^4

c

8y3−36xy2+54x2y−27x38y^3 - 36xy^2 + 54x^2y - 27x^3

Step-by-step solution

9 steps

Miriam, wacha tuendelee na upanuzi wa binomial.

Part a· 3 steps

Panua (m−2n)5(m-2n)^5

Tunatumia Theorem ya Binomial, ambayo inasema (a+b)N=∑k=0N(Nk)aN−kbk(a+b)^N = \sum_{k=0}^{N} \binom{N}{k} a^{N-k} b^k.
Kwa (m−2n)5(m-2n)^5, tuna a=ma=m, b=−2nb=-2n, na N=5N=5.
Viwianishi vya binomial kwa N=5N=5 ni 1,5,10,10,5,11, 5, 10, 10, 5, 1.

  1. Andika maneno kwa kutumia viwianishi vya binomial na nguvu za mm na (−2n)(-2n).

    (m−2n)5=(50)m5(−2n)0+(51)m4(−2n)1+(52)m3(−2n)2+(53)m2(−2n)3+(54)m1(−2n)4+(55)m0(−2n)5(m-2n)^5 = \binom{5}{0}m^5(-2n)^0 + \binom{5}{1}m^4(-2n)^1 + \binom{5}{2}m^3(-2n)^2 + \binom{5}{3}m^2(-2n)^3 + \binom{5}{4}m^1(-2n)^4 + \binom{5}{5}m^0(-2n)^5

  2. Rahisisha kila neno.

    =1⋅m5⋅1+5⋅m4⋅(−2n)+10⋅m3⋅(4n2)+10⋅m2⋅(−8n3)+5⋅m⋅(16n4)+1⋅1⋅(−32n5)= 1 \cdot m^5 \cdot 1 + 5 \cdot m^4 \cdot (-2n) + 10 \cdot m^3 \cdot (4n^2) + 10 \cdot m^2 \cdot (-8n^3) + 5 \cdot m \cdot (16n^4) + 1 \cdot 1 \cdot (-32n^5)

  3. Fanya kuzidisha.

    =m5−10m4n+40m3n2−80m2n3+80mn4−32n5= m^5 - 10m^4n + 40m^3n^2 - 80m^2n^3 + 80mn^4 - 32n^5

    Upanuzi kamili ni:

    m5−10m4n+40m3n2−80m2n3+80mn4−32n5\boxed{m^5 - 10m^4n + 40m^3n^2 - 80m^2n^3 + 80mn^4 - 32n^5}

Part b

81x4+108x3y+54x2y2+12xy3+y481x^4 + 108x^3y + 54x^2y^2 + 12xy^3 + y^4

Panua (3x+y)4(3x+y)^4

Hapa tuna a=3xa=3x, b=yb=y, na N=4N=4.
Viwianishi vya binomial kwa N=4N=4 ni 1,4,6,4,11, 4, 6, 4, 1.

  1. Andika maneno kwa kutumia viwianishi vya binomial na nguvu za 3x3x na yy.

    (3x+y)4=(40)(3x)4y0+(41)(3x)3y1+(42)(3x)2y2+(43)(3x)1y3+(44)(3x)0y4(3x+y)^4 = \binom{4}{0}(3x)^4y^0 + \binom{4}{1}(3x)^3y^1 + \binom{4}{2}(3x)^2y^2 + \binom{4}{3}(3x)^1y^3 + \binom{4}{4}(3x)^0y^4

  2. Rahisisha kila neno.

    =1⋅(81x4)⋅1+4⋅(27x3)⋅y+6⋅(9x2)⋅y2+4⋅(3x)⋅y3+1⋅1⋅y4= 1 \cdot (81x^4) \cdot 1 + 4 \cdot (27x^3) \cdot y + 6 \cdot (9x^2) \cdot y^2 + 4 \cdot (3x) \cdot y^3 + 1 \cdot 1 \cdot y^4

  3. Fanya kuzidisha.

    =81x4+108x3y+54x2y2+12xy3+y4= 81x^4 + 108x^3y + 54x^2y^2 + 12xy^3 + y^4

    Upanuzi kamili ni:

    81x4+108x3y+54x2y2+12xy3+y4\boxed{81x^4 + 108x^3y + 54x^2y^2 + 12xy^3 + y^4}

Part c

8y3−36xy2+54x2y−27x38y^3 - 36xy^2 + 54x^2y - 27x^3

Panua (2y−3x)3(2y-3x)^3

Hapa tuna a=2ya=2y, b=−3xb=-3x, na N=3N=3.
Viwianishi vya binomial kwa N=3N=3 ni 1,3,3,11, 3, 3, 1.

  1. Andika maneno kwa kutumia viwianishi vya binomial na nguvu za 2y2y na (−3x)(-3x).

    (2y−3x)3=(30)(2y)3(−3x)0+(31)(2y)2(−3x)1+(32)(2y)1(−3x)2+(33)(2y)0(−3x)3(2y-3x)^3 = \binom{3}{0}(2y)^3(-3x)^0 + \binom{3}{1}(2y)^2(-3x)^1 + \binom{3}{2}(2y)^1(-3x)^2 + \binom{3}{3}(2y)^0(-3x)^3

  2. Rahisisha kila neno.

    =1⋅(8y3)⋅1+3⋅(4y2)⋅(−3x)+3⋅(2y)⋅(9x2)+1⋅1⋅(−27x3)= 1 \cdot (8y^3) \cdot 1 + 3 \cdot (4y^2) \cdot (-3x) + 3 \cdot (2y) \cdot (9x^2) + 1 \cdot 1 \cdot (-27x^3)

  3. Fanya kuzidisha.

    =8y3−36xy2+54x2y−27x3= 8y^3 - 36xy^2 + 54x^2y - 27x^3

    Upanuzi kamili ni:

    8y3−36xy2+54x2y−27x3\boxed{8y^3 - 36xy^2 + 54x^2y - 27x^3}
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