Question
Answer
$$(x+13)^3(x+19)^3=x^6+96x^5+3813x^4+80192x^3+941811x^2+5856864x+15069223$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+13)^3(x+19)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^3+39x^2+507x+2197)(x^3+57x^2+1083x+6859) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^6+96x^5+3813x^4+80192x^3+941811x^2+5856864x+15069223\end{aligned} $$ | |
| ① | Find $ \left(x+13\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 13 $. $$ \left(x+13\right)^3 = x^3+3 \cdot x^2 \cdot 13 + 3 \cdot x \cdot 13^2+13^3 = x^3+39x^2+507x+2197 $$Find $ \left(x+19\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = x $ and $ B = 19 $. $$ \left(x+19\right)^3 = x^3+3 \cdot x^2 \cdot 19 + 3 \cdot x \cdot 19^2+19^3 = x^3+57x^2+1083x+6859 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^3+39x^2+507x+2197}\right) $ by each term in $ \left( x^3+57x^2+1083x+6859\right) $. $$ \left( \color{blue}{x^3+39x^2+507x+2197}\right) \cdot \left( x^3+57x^2+1083x+6859\right) = \\ = x^6+57x^5+1083x^4+6859x^3+39x^5+2223x^4+42237x^3+267501x^2+507x^4+28899x^3+549081x^2+3477513x+2197x^3+125229x^2+2379351x+15069223 $$ |
| ③ | Combine like terms: $$ x^6+ \color{blue}{57x^5} + \color{red}{1083x^4} + \color{green}{6859x^3} + \color{blue}{39x^5} + \color{orange}{2223x^4} + \color{blue}{42237x^3} + \color{red}{267501x^2} + \color{orange}{507x^4} + \color{green}{28899x^3} + \color{orange}{549081x^2} + \color{blue}{3477513x} + \color{green}{2197x^3} + \color{orange}{125229x^2} + \color{blue}{2379351x} +15069223 = \\ = x^6+ \color{blue}{96x^5} + \color{orange}{3813x^4} + \color{green}{80192x^3} + \color{orange}{941811x^2} + \color{blue}{5856864x} +15069223 $$ |
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