Question
Answer
$$(x+3)^3-15(x+3)^3+91(x+3)^2-249(x+3)^2+252=-14x^3-284x^2-1326x-1548$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+3)^3-15(x+3)^3+91(x+3)^2-249(x+3)^2+252& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3+9x^2+27x+27-15(x^3+9x^2+27x+27)+91(x^2+6x+9)-249(x^2+6x+9)+252 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3+9x^2+27x+27-(15x^3+135x^2+405x+405)+91x^2+546x+819-(249x^2+1494x+2241)+252 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+9x^2+27x+27-15x^3-135x^2-405x-405+91x^2+546x+819-(249x^2+1494x+2241)+252 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-14x^3-126x^2-378x-378+91x^2+546x+819-(249x^2+1494x+2241)+252 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-14x^3-35x^2+168x+441-(249x^2+1494x+2241)+252 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}-14x^3-35x^2+168x+441-249x^2-1494x-2241+252 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-14x^3-284x^2-1326x-1548\end{aligned} $$ | |
| ① | Find $ \left(x+3\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 3 $. $$ \left(x+3\right)^3 = x^3+3 \cdot x^2 \cdot 3 + 3 \cdot x \cdot 3^2+3^3 = x^3+9x^2+27x+27 $$Find $ \left(x+3\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 3 $. $$ \left(x+3\right)^3 = x^3+3 \cdot x^2 \cdot 3 + 3 \cdot x \cdot 3^2+3^3 = x^3+9x^2+27x+27 $$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 }$. $$ \begin{aligned}\left(x+3\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 3 + \color{red}{3^2} = x^2+6x+9\end{aligned} $$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 $ \color{blue}{15} $ by $ \left( x^3+9x^2+27x+27\right) $ $$ \color{blue}{15} \cdot \left( x^3+9x^2+27x+27\right) = 15x^3+135x^2+405x+405 $$Multiply $ \color{blue}{91} $ by $ \left( x^2+6x+9\right) $ $$ \color{blue}{91} \cdot \left( x^2+6x+9\right) = 91x^2+546x+819 $$Multiply $ \color{blue}{249} $ by $ \left( x^2+6x+9\right) $ $$ \color{blue}{249} \cdot \left( x^2+6x+9\right) = 249x^2+1494x+2241 $$ |
| ③ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 15x^3+135x^2+405x+405 \right) = -15x^3-135x^2-405x-405 $$ |
| ④ | Combine like terms: $$ \color{blue}{x^3} + \color{red}{9x^2} + \color{green}{27x} + \color{orange}{27} \color{blue}{-15x^3} \color{red}{-135x^2} \color{green}{-405x} \color{orange}{-405} = \\ = \color{blue}{-14x^3} \color{red}{-126x^2} \color{green}{-378x} \color{orange}{-378} $$ |
| ⑤ | Combine like terms: $$ -14x^3 \color{blue}{-126x^2} \color{red}{-378x} \color{green}{-378} + \color{blue}{91x^2} + \color{red}{546x} + \color{green}{819} = -14x^3 \color{blue}{-35x^2} + \color{red}{168x} + \color{green}{441} $$ |
| ⑥ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 249x^2+1494x+2241 \right) = -249x^2-1494x-2241 $$ |
| ⑦ | Combine like terms: $$ -14x^3 \color{blue}{-35x^2} + \color{red}{168x} + \color{green}{441} \color{blue}{-249x^2} \color{red}{-1494x} \color{orange}{-2241} + \color{orange}{252} = \\ = -14x^3 \color{blue}{-284x^2} \color{red}{-1326x} \color{orange}{-1548} $$ |
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