Question
Answer
$$(x+1)^2(x-3)^2=x^4-4x^3-2x^2+12x+9$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+1)^2(x-3)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+2x+1)(x^2-6x+9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^4-4x^3-2x^2+12x+9\end{aligned} $$ | |
| 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 }$. $$ \begin{aligned}\left(x+1\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 1 + \color{red}{1^2} = x^2+2x+1\end{aligned} $$Find $ \left(x-3\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}{ 3 }$. $$ \begin{aligned}\left(x-3\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 3 + \color{red}{3^2} = x^2-6x+9\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2+2x+1}\right) $ by each term in $ \left( x^2-6x+9\right) $. $$ \left( \color{blue}{x^2+2x+1}\right) \cdot \left( x^2-6x+9\right) = x^4-6x^3+9x^2+2x^3-12x^2+18x+x^2-6x+9 $$ |
| ③ | Combine like terms: $$ x^4 \color{blue}{-6x^3} + \color{red}{9x^2} + \color{blue}{2x^3} \color{green}{-12x^2} + \color{orange}{18x} + \color{green}{x^2} \color{orange}{-6x} +9 = x^4 \color{blue}{-4x^3} \color{green}{-2x^2} + \color{orange}{12x} +9 $$ |
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