Question

Resolva:
Qual é o desenvolvimento do binômio (a-3) ³

Qual é o desenvolvimento do binômio (a-5) ²

Qual é o desenvolvimento do binômio (a-8) ³

Answer 1

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$\boxed{\underline{\sf cubo~da~diferenc_{\!\!,}a~de~dois~termos}}\\\boxed{\boxed{\boxed{\boxed{\sf(a-b)^3=a^3-3\cdot a^2\cdot b+3\cdot a\cdot b^2-b^3 }}}}\\\boxed{\underline{\sf quadrado~da~diferenc_{\!\!,}a~de~dois~termos}}\\\boxed{\boxed{\boxed{\boxed{\sf (a-b)^2=a^2-2ab+b^2 }}}}\\\boxed{\underline{\sf bin\hat omio~de~Newton}}\\\boxed{\boxed{\boxed{\boxed{\sf(a+b)^n=\binom{n}{0}a^n+\binom{n}{1}a^{n-1}b^1+\binom{n}{2}a^{n-2}b^2+...}}}}$

$\tt a)~\underline{\sf por~bin\hat omio~de~Newton:}\\\sf (a-3)^3=\binom{3}{0}a^3-\binom{3}{1}a^{3-1}\cdot3+\binom{3}{2}a^{3-2}\cdot3^2-\binom{3}{3}a^{3-3}\cdot3^3\\\sf \binom{3}{0}=1~\binom{3}{1}=3\\\sf \binom{3}{2}=\dfrac{3\cdot2}{2\cdot1}=3\\\sf \binom{3}{3}=1\\\sf (a-3)^3=a^3-3a^2\cdot3+3\cdot a\cdot9-27\\\sf(a-3)^3=a^3-9a^2+27a-27\\\underline{\sf pelo~cubo~da~diferenc_{\!\!,}a~de~dois~termos:}\\\sf (a-3)^3=a^3-3\cdot a^2\cdot3+3\cdot a\cdot3^2-3^3\\\sf (a-3)^3=a^3-9a^2+27a-27$

$\tt b)~\underline{\sf por~bin\hat omio~de~Newton:}\\\sf (a-5)^2=\binom{2}{0}a^2+\binom{2}{1}a^{2-1}\cdot 5+\binom{2}{2}a^{2-2}5^2\\\sf\binom{2}{0}=1~\binom{2}{1}=2~~\binom{2}{2}=1\\\sf (a-5)^2=1a^2-2\cdot a\cdot5+1\cdot 5^2\\\sf (a-5)^2=a^2-10a+25\\\underline{\sf pelo~quadrado~diferenc_{\!\!,}a~de~dois~termos:}\\\sf(a-5)^2=a^2-2\cdot a\cdot 5+5^2\\\sf(a-5)^2=a^2-10a+25$

$\tt c)~\underline{\sf por~bin\hat omio~de~Newton:}\\\sf (a-8)^3=\binom{3}{0}a^3-\binom{3}{1}a^{3-1}8+\binom{3}{2}a^{3-2}8^2-\binom{3}{3}a^{3-3}8^3\\\sf\binom{3}{0}=1~\binom{3}{1}=3~\binom{3}{2}=3~\binom{3}{3}=1\\\sf (a-8)^3=1a^3-3\cdot a^2\cdot8+3\cdot a\cdot8^2-1\cdot8^3\\\sf (a-8)^3=a^3-24a^2+192a-512\\\underline{\sf pelo~cubo~da~diferenc_{\!\!,}a~de~dois~termos:}\\\sf (a-8)^3=a^3-3\cdot a^2\cdot 8+3\cdot a\cdot 8^3-8^3\\\sf (a-8)^3=a^3-24a^2+192a-512$