Question
Answer
$$((y-5)^2-2)^2-y+3=y^4-20y^3+146y^2-461y+532$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}((y-5)^2-2)^2-y+3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(1y^2-10y+25-2)^2-y+3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(1y^2-10y+23)^2-y+3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}y^4-20y^3+146y^2-460y+529-y+3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}y^4-20y^3+146y^2-461y+532\end{aligned} $$ | |
| ① | Find $ \left(y-5\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ y } $ and $ B = \color{red}{ 5 }$. $$ \begin{aligned}\left(y-5\right)^2 = \color{blue}{y^2} -2 \cdot y \cdot 5 + \color{red}{5^2} = y^2-10y+25\end{aligned} $$ |
| ② | Combine like terms: $$ y^2-10y+ \color{blue}{25} \color{blue}{-2} = y^2-10y+ \color{blue}{23} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{y^2-10y+23}\right) $ by each term in $ \left( y^2-10y+23\right) $. $$ \left( \color{blue}{y^2-10y+23}\right) \cdot \left( y^2-10y+23\right) = y^4-10y^3+23y^2-10y^3+100y^2-230y+23y^2-230y+529 $$ |
| ④ | Combine like terms: $$ y^4 \color{blue}{-10y^3} + \color{red}{23y^2} \color{blue}{-10y^3} + \color{green}{100y^2} \color{orange}{-230y} + \color{green}{23y^2} \color{orange}{-230y} +529 = \\ = y^4 \color{blue}{-20y^3} + \color{green}{146y^2} \color{orange}{-460y} +529 $$ |
| ⑤ | Combine like terms: $$ y^4-20y^3+146y^2 \color{blue}{-460y} + \color{red}{529} \color{blue}{-y} + \color{red}{3} = y^4-20y^3+146y^2 \color{blue}{-461y} + \color{red}{532} $$ |
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