Question
Answer
$$(2x)^3+(2x+1)^3+(2k+2)^3=8k^3+16x^3+24k^2+12x^2+24k+6x+9$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(2x)^3+(2x+1)^3+(2k+2)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}8x^3+8x^3+12x^2+6x+1+8k^3+24k^2+24k+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}16x^3+12x^2+6x+1+8k^3+24k^2+24k+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}8k^3+16x^3+24k^2+12x^2+24k+6x+9\end{aligned} $$ | |
| ① | $$ \left( 2x \right)^3 = 2^3x^3 = 8x^3 $$ Find $ \left(2x+1\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2x $ and $ B = 1 $. $$ \left(2x+1\right)^3 = \left( 2x \right)^3+3 \cdot \left( 2x \right)^2 \cdot 1 + 3 \cdot 2x \cdot 1^2+1^3 = 8x^3+12x^2+6x+1 $$Find $ \left(2k+2\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 2k $ and $ B = 2 $. $$ \left(2k+2\right)^3 = \left( 2k \right)^3+3 \cdot \left( 2k \right)^2 \cdot 2 + 3 \cdot 2k \cdot 2^2+2^3 = 8k^3+24k^2+24k+8 $$ |
| ② | Combine like terms: $$ \color{blue}{8x^3} + \color{blue}{8x^3} +12x^2+6x+1 = \color{blue}{16x^3} +12x^2+6x+1 $$ |
| ③ | Combine like terms: $$ 16x^3+12x^2+6x+ \color{blue}{1} +8k^3+24k^2+24k+ \color{blue}{8} = 8k^3+16x^3+24k^2+12x^2+24k+6x+ \color{blue}{9} $$ |
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