Question
Answer
$$(x-1)^4+8(x-1)^3+24(x-1)^2+32(x-1)+16=x^4+4x^3+6x^2+4x+1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-1)^4+8(x-1)^3+24(x-1)^2+32(x-1)+16& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^4-4x^3+6x^2-4x+1+8(x^3-3x^2+3x-1)+24(x^2-2x+1)+32(x-1)+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-4x^3+6x^2-4x+1+8x^3-24x^2+24x-8+24x^2-48x+24+32x-32+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^4+4x^3-18x^2+20x-7+24x^2-48x+24+32x-32+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^4+4x^3+6x^2-28x+17+32x-32+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^4+4x^3+6x^2+4x-15+16 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}x^4+4x^3+6x^2+4x+1\end{aligned} $$ | |
| ① | $$ (x-1)^4 = (x-1)^2 \cdot (x-1)^2 $$ |
| ② | Find $ \left(x-1\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}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-2x+1}\right) $ by each term in $ \left( x^2-2x+1\right) $. $$ \left( \color{blue}{x^2-2x+1}\right) \cdot \left( x^2-2x+1\right) = x^4-2x^3+x^2-2x^3+4x^2-2x+x^2-2x+1 $$ |
| ④ | Combine like terms: $$ x^4 \color{blue}{-2x^3} + \color{red}{x^2} \color{blue}{-2x^3} + \color{green}{4x^2} \color{orange}{-2x} + \color{green}{x^2} \color{orange}{-2x} +1 = x^4 \color{blue}{-4x^3} + \color{green}{6x^2} \color{orange}{-4x} +1 $$Find $ \left(x-1\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = x $ and $ B = 1 $. $$ \left(x-1\right)^3 = x^3-3 \cdot x^2 \cdot 1 + 3 \cdot x \cdot 1^2-1^3 = x^3-3x^2+3x-1 $$Find $ \left(x-1\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}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$ |
| ⑤ | Multiply $ \color{blue}{8} $ by $ \left( x^3-3x^2+3x-1\right) $ $$ \color{blue}{8} \cdot \left( x^3-3x^2+3x-1\right) = 8x^3-24x^2+24x-8 $$Multiply $ \color{blue}{24} $ by $ \left( x^2-2x+1\right) $ $$ \color{blue}{24} \cdot \left( x^2-2x+1\right) = 24x^2-48x+24 $$Multiply $ \color{blue}{32} $ by $ \left( x-1\right) $ $$ \color{blue}{32} \cdot \left( x-1\right) = 32x-32 $$ |
| ⑥ | Combine like terms: $$ x^4 \color{blue}{-4x^3} + \color{red}{6x^2} \color{green}{-4x} + \color{orange}{1} + \color{blue}{8x^3} \color{red}{-24x^2} + \color{green}{24x} \color{orange}{-8} = \\ = x^4+ \color{blue}{4x^3} \color{red}{-18x^2} + \color{green}{20x} \color{orange}{-7} $$ |
| ⑦ | Combine like terms: $$ x^4+4x^3 \color{blue}{-18x^2} + \color{red}{20x} \color{green}{-7} + \color{blue}{24x^2} \color{red}{-48x} + \color{green}{24} = x^4+4x^3+ \color{blue}{6x^2} \color{red}{-28x} + \color{green}{17} $$ |
| ⑧ | Combine like terms: $$ x^4+4x^3+6x^2 \color{blue}{-28x} + \color{red}{17} + \color{blue}{32x} \color{red}{-32} = x^4+4x^3+6x^2+ \color{blue}{4x} \color{red}{-15} $$ |
| ⑨ | Combine like terms: $$ x^4+4x^3+6x^2+4x \color{blue}{-15} + \color{blue}{16} = x^4+4x^3+6x^2+4x+ \color{blue}{1} $$ |
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