Question
Answer
$$(x-1)^3+5.5(x-1)^2+10(x-1)+6=x^3+2x^2+3x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-1)^3+5.5(x-1)^2+10(x-1)+6& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3-3x^2+3x-1+5.5(x^2-2x+1)+10(x-1)+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3-3x^2+3x-1+5x^2-10x+5+10x-10+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+2x^2-7x+4+10x-10+6 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^3+2x^2+3x-6+6 \xlongequal{ } \\[1 em] & \xlongequal{ }x^3+2x^2+3x -\cancel{6}+ \cancel{6} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^3+2x^2+3x\end{aligned} $$ | |
| 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 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$ |
| ② | Multiply $ \color{blue}{5} $ by $ \left( x^2-2x+1\right) $ $$ \color{blue}{5} \cdot \left( x^2-2x+1\right) = 5x^2-10x+5 $$Multiply $ \color{blue}{10} $ by $ \left( x-1\right) $ $$ \color{blue}{10} \cdot \left( x-1\right) = 10x-10 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-3x^2} + \color{red}{3x} \color{green}{-1} + \color{blue}{5x^2} \color{red}{-10x} + \color{green}{5} = x^3+ \color{blue}{2x^2} \color{red}{-7x} + \color{green}{4} $$ |
| ④ | Combine like terms: $$ x^3+2x^2 \color{blue}{-7x} + \color{red}{4} + \color{blue}{10x} \color{red}{-10} = x^3+2x^2+ \color{blue}{3x} \color{red}{-6} $$ |
| ⑤ | Combine like terms: $$ x^3+2x^2+3x \, \color{blue}{ -\cancel{6}} \,+ \, \color{blue}{ \cancel{6}} \, = x^3+2x^2+3x $$ |
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