Question
Answer
$$(x+25)^4+264(25+x)^3+272(25+x)^2+67584\cdot(25+x)+16384=x^4+364x^3+23822x^2+638684x+6391609$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x+25)^4+264(25+x)^3+272(25+x)^2+67584\cdot(25+x)+16384& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^4+100x^3+3750x^2+62500x+390625+264(15625+1875x+75x^2+x^3)+272(625+50x+x^2)+67584\cdot(25+x)+16384 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4+100x^3+3750x^2+62500x+390625+4125000+495000x+19800x^2+264x^3+170000+13600x+272x^2+1689600+67584x+16384 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^4+364x^3+23550x^2+557500x+4515625+170000+13600x+272x^2+1689600+67584x+16384 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^4+364x^3+23822x^2+571100x+4685625+1689600+67584x+16384 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}x^4+364x^3+23822x^2+638684x+6375225+16384 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}x^4+364x^3+23822x^2+638684x+6391609\end{aligned} $$ | |
| ① | $$ (x+25)^4 = (x+25)^2 \cdot (x+25)^2 $$ |
| ② | Find $ \left(x+25\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}{ 25 }$. $$ \begin{aligned}\left(x+25\right)^2 = \color{blue}{x^2} +2 \cdot x \cdot 25 + \color{red}{25^2} = x^2+50x+625\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2+50x+625}\right) $ by each term in $ \left( x^2+50x+625\right) $. $$ \left( \color{blue}{x^2+50x+625}\right) \cdot \left( x^2+50x+625\right) = \\ = x^4+50x^3+625x^2+50x^3+2500x^2+31250x+625x^2+31250x+390625 $$ |
| ④ | Combine like terms: $$ x^4+ \color{blue}{50x^3} + \color{red}{625x^2} + \color{blue}{50x^3} + \color{green}{2500x^2} + \color{orange}{31250x} + \color{green}{625x^2} + \color{orange}{31250x} +390625 = \\ = x^4+ \color{blue}{100x^3} + \color{green}{3750x^2} + \color{orange}{62500x} +390625 $$Find $ \left(25+x\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 25 $ and $ B = x $. $$ \left(25+x\right)^3 = 25^3+3 \cdot 25^2 \cdot x + 3 \cdot 25 \cdot x^2+x^3 = 15625+1875x+75x^2+x^3 $$Find $ \left(25+x\right)^2 $ using formula. $$ (A + B)^2 = \color{blue}{A^2} + 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 25 } $ and $ B = \color{red}{ x }$. $$ \begin{aligned}\left(25+x\right)^2 = \color{blue}{25^2} +2 \cdot 25 \cdot x + \color{red}{x^2} = 625+50x+x^2\end{aligned} $$ |
| ⑤ | Multiply $ \color{blue}{264} $ by $ \left( 15625+1875x+75x^2+x^3\right) $ $$ \color{blue}{264} \cdot \left( 15625+1875x+75x^2+x^3\right) = 4125000+495000x+19800x^2+264x^3 $$Multiply $ \color{blue}{272} $ by $ \left( 625+50x+x^2\right) $ $$ \color{blue}{272} \cdot \left( 625+50x+x^2\right) = 170000+13600x+272x^2 $$Multiply $ \color{blue}{67584} $ by $ \left( 25+x\right) $ $$ \color{blue}{67584} \cdot \left( 25+x\right) = 1689600+67584x $$ |
| ⑥ | Combine like terms: $$ x^4+ \color{blue}{100x^3} + \color{red}{3750x^2} + \color{green}{62500x} + \color{orange}{390625} + \color{orange}{4125000} + \color{green}{495000x} + \color{red}{19800x^2} + \color{blue}{264x^3} = \\ = x^4+ \color{blue}{364x^3} + \color{red}{23550x^2} + \color{green}{557500x} + \color{orange}{4515625} $$ |
| ⑦ | Combine like terms: $$ x^4+364x^3+ \color{blue}{23550x^2} + \color{red}{557500x} + \color{green}{4515625} + \color{green}{170000} + \color{red}{13600x} + \color{blue}{272x^2} = \\ = x^4+364x^3+ \color{blue}{23822x^2} + \color{red}{571100x} + \color{green}{4685625} $$ |
| ⑧ | Combine like terms: $$ x^4+364x^3+23822x^2+ \color{blue}{571100x} + \color{red}{4685625} + \color{red}{1689600} + \color{blue}{67584x} = x^4+364x^3+23822x^2+ \color{blue}{638684x} + \color{red}{6375225} $$ |
| ⑨ | Combine like terms: $$ x^4+364x^3+23822x^2+638684x+ \color{blue}{6375225} + \color{blue}{16384} = x^4+364x^3+23822x^2+638684x+ \color{blue}{6391609} $$ |
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