Đáp án:
\(\begin{array}{l}
3,\\
a,\\
A = {\left( {x - 2} \right)^3}\\
x = 12 \Rightarrow A = 1000\\
b,\\
B = {\left( {2x + 1} \right)^3}\\
x = 5 \Rightarrow B = 1331\\
4,\\
a,\\
1061200\\
b,\\
117650\\
c,\\
4940\\
d,\\
216000\\
e,\\
125000000\\
5,\\
a,\\
x = - 2\\
b,\\
x = - 3\\
c,\\
x = - \dfrac{3}{2}
\end{array}\)
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
3,\\
a,\\
A = {x^3} - 6{x^2} + 12x - 8\\
= {x^3} - 3.2.{x^2} + 3.4.x - 8\\
= {x^3} - 3.{x^2}.2 + 3.x{.2^2} - {2^3}\\
= {\left( {x - 2} \right)^3}\\
x = 12 \Rightarrow A = {\left( {12 - 2} \right)^3} = {10^3} = 1000\\
b,\\
B = {\left( {2x - 1} \right)^3} + 6.{\left( {2x - 1} \right)^2} + 12.\left( {2x - 1} \right) + 8\\
= {\left( {2x - 1} \right)^3} + 3.2.{\left( {2x - 1} \right)^2} + 3.4.\left( {2x - 1} \right) + 8\\
= {\left( {2x - 1} \right)^3} + 3.{\left( {2x - 1} \right)^2}.2 + 3.\left( {2x - 1} \right){.2^2} + {2^3}\\
= {\left[ {\left( {2x - 1} \right) + 2} \right]^3}\\
= {\left( {2x + 1} \right)^3}\\
x = 5 \Rightarrow B = {\left( {2.5 + 1} \right)^3} = {11^3} = 1331\\
4,\\
a,\\
{102^3} - {2^3}\\
= \left( {{{102}^3} - {{3.102}^2}.2 + {{3.102.2}^2} - {2^3}} \right) + {3.102^2}.2 - {3.102.2^2}\\
= {\left( {102 - 3} \right)^3} + 3.102.2.\left( {102 - 2} \right)\\
= {100^3} + 306.2.100\\
= 1000000 + 61200\\
= 1061200\\
b,\\
{49^3} + 1\\
= {49^3} + {1^3}\\
= \left( {{{49}^3} + {{3.49}^2}.1 + {{3.49.1}^2} + {1^3}} \right) - {3.49^2}.1 - {3.49.1^2}\\
= {\left( {49 + 1} \right)^3} - 3.49.1.\left( {49 + 1} \right)\\
= {50^3} - 147.50\\
= 125000 - 7350\\
= 117650\\
c,\\
{17^3} + {3^3}\\
= \left( {{{17}^3} + {{3.17}^2}.3 + {{3.17.3}^2} + {3^3}} \right) - \left( {{{3.17}^2}.3 + {{3.17.3}^2}} \right)\\
= {\left( {17 + 3} \right)^3} - 3.17.3.\left( {17 + 3} \right)\\
= {20^3} - 51.3.20\\
= 8000 - 153.20\\
= 8000 - 3060\\
= 4940\\
d,\\
{59^3} + {3.59^2} + 3.59 + 1\\
= {59^3} + {3.59^2}.1 + {3.59.1^2} + {1^3}\\
= {\left( {59 + 1} \right)^3}\\
= {60^3}\\
= 216000\\
e,\\
{503^3} - {9.503^2} + 27.503 - 27\\
= {503^3} - {3.3.503^2} + 3.9.503 - 27\\
= {503^3} - {3.503^2}.3 + {3.503.3^2} - {3^3}\\
= {\left( {503 - 3} \right)^3}\\
= {500^3}\\
= 125000000\\
5,\\
a,\\
{x^3} + 3{x^2} + 3x + 1 = {\left( {3x + 5} \right)^3}\\
\Leftrightarrow {x^3} + 3.{x^2}.1 + 3.x{.1^2} + {1^3} = {\left( {3x + 5} \right)^3}\\
\Leftrightarrow {\left( {x + 1} \right)^3} = {\left( {3x + 5} \right)^3}\\
\Leftrightarrow x + 1 = 3x + 5\\
\Leftrightarrow 3x + 5 - x - 1 = 0\\
\Leftrightarrow 2x + 4 = 0\\
\Leftrightarrow x = - 2\\
b,\\
{x^3} - 6{x^2} + 12x - 8 = {\left( {2x + 1} \right)^3}\\
\Leftrightarrow {x^3} - 3.2.{x^2} + 3.4.x - 8 = {\left( {2x + 1} \right)^3}\\
\Leftrightarrow {x^3} - 3.{x^2}.2 + 3.x{.2^2} - {2^3} = {\left( {2x + 1} \right)^3}\\
\Leftrightarrow {\left( {x - 2} \right)^3} = {\left( {2x + 1} \right)^3}\\
\Leftrightarrow x - 2 = 2x + 1\\
\Leftrightarrow 2x + 1 - x + 2 = 0\\
\Leftrightarrow x + 3 = 0\\
\Leftrightarrow x = - 3\\
c,\\
8{x^3} + 36{x^2} + 54x + 27 = 0\\
\Leftrightarrow 8{x^3} + 3.3.4{x^2} + 3.2.9x + 27 = 0\\
\Leftrightarrow {\left( {2x} \right)^3} + 3.{\left( {2x} \right)^2}.3 + 3.2x{.3^2} + {3^3} = 0\\
\Leftrightarrow {\left( {2x + 3} \right)^3} = 0\\
\Leftrightarrow 2x + 3 = 0\\
\Leftrightarrow x = - \dfrac{3}{2}
\end{array}\)
