Question
Answer
$$x^3+(x+1)^3+(x+2)^3=3x^3+9x^2+15x+9$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x^3+(x+1)^3+(x+2)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3+x^3+3x^2+3x+1+x^3+6x^2+12x+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}2x^3+3x^2+3x+1+x^3+6x^2+12x+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}3x^3+9x^2+15x+9\end{aligned} $$ | |
| 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+2\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 2 $. $$ \left(x+2\right)^3 = x^3+3 \cdot x^2 \cdot 2 + 3 \cdot x \cdot 2^2+2^3 = x^3+6x^2+12x+8 $$ |
| ② | Combine like terms: $$ \color{blue}{x^3} + \color{blue}{x^3} +3x^2+3x+1 = \color{blue}{2x^3} +3x^2+3x+1 $$ |
| ③ | Combine like terms: $$ \color{blue}{2x^3} + \color{red}{3x^2} + \color{green}{3x} + \color{orange}{1} + \color{blue}{x^3} + \color{red}{6x^2} + \color{green}{12x} + \color{orange}{8} = \\ = \color{blue}{3x^3} + \color{red}{9x^2} + \color{green}{15x} + \color{orange}{9} $$ |
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