Question
Answer
$$(3+x)^3+(4+x)^3+(5+x)^3=3x^3+36x^2+150x+216$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(3+x)^3+(4+x)^3+(5+x)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}27+27x+9x^2+x^3+64+48x+12x^2+x^3+125+75x+15x^2+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}2x^3+21x^2+75x+91+125+75x+15x^2+x^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}3x^3+36x^2+150x+216\end{aligned} $$ | |
| ① | Find $ \left(3+x\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 3 $ and $ B = x $. $$ \left(3+x\right)^3 = 3^3+3 \cdot 3^2 \cdot x + 3 \cdot 3 \cdot x^2+x^3 = 27+27x+9x^2+x^3 $$Find $ \left(4+x\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 4 $ and $ B = x $. $$ \left(4+x\right)^3 = 4^3+3 \cdot 4^2 \cdot x + 3 \cdot 4 \cdot x^2+x^3 = 64+48x+12x^2+x^3 $$Find $ \left(5+x\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 5 $ and $ B = x $. $$ \left(5+x\right)^3 = 5^3+3 \cdot 5^2 \cdot x + 3 \cdot 5 \cdot x^2+x^3 = 125+75x+15x^2+x^3 $$ |
| ② | Combine like terms: $$ \color{blue}{27} + \color{red}{27x} + \color{green}{9x^2} + \color{orange}{x^3} + \color{blue}{64} + \color{red}{48x} + \color{green}{12x^2} + \color{orange}{x^3} = \\ = \color{orange}{2x^3} + \color{green}{21x^2} + \color{red}{75x} + \color{blue}{91} $$ |
| ③ | Combine like terms: $$ \color{blue}{2x^3} + \color{red}{21x^2} + \color{green}{75x} + \color{orange}{91} + \color{orange}{125} + \color{green}{75x} + \color{red}{15x^2} + \color{blue}{x^3} = \\ = \color{blue}{3x^3} + \color{red}{36x^2} + \color{green}{150x} + \color{orange}{216} $$ |
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