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

Miriam, wacha tuendelee na upanuzi wa binomial.

a) Panua $(m-2n)^5$

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

Step 1: Andika maneno kwa kutumia viwianishi vya binomial na nguvu za $m$ na $(-2n)$. $(m-2n)^5 = \binom{5}{0}m^5(-2n)^0 + \binom{5}{1}m^4(-2n)^1 + \binom{5}{2}m^3(-2n)^2 + \binom{5}{3}m^2(-2n)^3 + \binom{5}{4}m^1(-2n)^4 + \binom{5}{5}m^0(-2n)^5$

Step 2: Rahisisha kila neno. $= 1 \cdot m^5 \cdot 1 + 5 \cdot m^4 \cdot (-2n) + 10 \cdot m^3 \cdot (4n^2) + 10 \cdot m^2 \cdot (-8n^3) + 5 \cdot m \cdot (16n^4) + 1 \cdot 1 \cdot (-32n^5)$

Step 3: Fanya kuzidisha. $= m^5 - 10m^4n + 40m^3n^2 - 80m^2n^3 + 80mn^4 - 32n^5$ Upanuzi kamili ni: $\boxed{m^5 - 10m^4n + 40m^3n^2 - 80m^2n^3 + 80mn^4 - 32n^5}$

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

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

Step 1: Andika maneno kwa kutumia viwianishi vya binomial na nguvu za $3x$ na $y$. $(3x+y)^4 = \binom{4}{0}(3x)^4y^0 + \binom{4}{1}(3x)^3y^1 + \binom{4}{2}(3x)^2y^2 + \binom{4}{3}(3x)^1y^3 + \binom{4}{4}(3x)^0y^4$

Step 2: Rahisisha kila neno. $= 1 \cdot (81x^4) \cdot 1 + 4 \cdot (27x^3) \cdot y + 6 \cdot (9x^2) \cdot y^2 + 4 \cdot (3x) \cdot y^3 + 1 \cdot 1 \cdot y^4$

Step 3: Fanya kuzidisha. $= 81x^4 + 108x^3y + 54x^2y^2 + 12xy^3 + y^4$ Upanuzi kamili ni: $\boxed{81x^4 + 108x^3y + 54x^2y^2 + 12xy^3 + y^4}$

c) Panua $(2y-3x)^3$

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

Step 1: Andika maneno kwa kutumia viwianishi vya binomial na nguvu za $2y$ na $(-3x)$. $(2y-3x)^3 = \binom{3}{0}(2y)^3(-3x)^0 + \binom{3}{1}(2y)^2(-3x)^1 + \binom{3}{2}(2y)^1(-3x)^2 + \binom{3}{3}(2y)^0(-3x)^3$

Step 2: Rahisisha kila neno. $= 1 \cdot (8y^3) \cdot 1 + 3 \cdot (4y^2) \cdot (-3x) + 3 \cdot (2y) \cdot (9x^2) + 1 \cdot 1 \cdot (-27x^3)$

Step 3: Fanya kuzidisha. $= 8y^3 - 36xy^2 + 54x^2y - 27x^3$ Upanuzi kamili ni: $\boxed{8y^3 - 36xy^2 + 54x^2y - 27x^3}$

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