Question
Answer
$$(x-4)^3+4(x-4)^2+4(x-4)+4=x^3-8x^2+20x-12$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-4)^3+4(x-4)^2+4(x-4)+4& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3-12x^2+48x-64+4(x^2-8x+16)+4(x-4)+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3-12x^2+48x-64+4x^2-32x+64+4x-16+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3-8x^2+16x+4x-16+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^3-8x^2+20x-16+4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^3-8x^2+20x-12\end{aligned} $$ | |
| ① | Find $ \left(x-4\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = x $ and $ B = 4 $. $$ \left(x-4\right)^3 = x^3-3 \cdot x^2 \cdot 4 + 3 \cdot x \cdot 4^2-4^3 = x^3-12x^2+48x-64 $$Find $ \left(x-4\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 4 }$. $$ \begin{aligned}\left(x-4\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 4 + \color{red}{4^2} = x^2-8x+16\end{aligned} $$ |
| ② | Multiply $ \color{blue}{4} $ by $ \left( x^2-8x+16\right) $ $$ \color{blue}{4} \cdot \left( x^2-8x+16\right) = 4x^2-32x+64 $$Multiply $ \color{blue}{4} $ by $ \left( x-4\right) $ $$ \color{blue}{4} \cdot \left( x-4\right) = 4x-16 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-12x^2} + \color{red}{48x} \, \color{green}{ -\cancel{64}} \,+ \color{blue}{4x^2} \color{red}{-32x} + \, \color{green}{ \cancel{64}} \, = x^3 \color{blue}{-8x^2} + \color{red}{16x} $$ |
| ④ | Combine like terms: $$ x^3-8x^2+ \color{blue}{16x} + \color{blue}{4x} -16 = x^3-8x^2+ \color{blue}{20x} -16 $$ |
| ⑤ | Combine like terms: $$ x^3-8x^2+20x \color{blue}{-16} + \color{blue}{4} = x^3-8x^2+20x \color{blue}{-12} $$ |
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