Question
Answer
$$(1-p)^4+p^4+4\cdot(1-p)p^3+4(1-p)^3p+6(1-p)^2p^2=1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(1-p)^4+p^4+4\cdot(1-p)p^3+4(1-p)^3p+6(1-p)^2p^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}p^4-4p^3+6p^2-4p+1+p^4+4\cdot(1-p)p^3+4(1-3p+3p^2-p^3)p+6(1-2p+p^2)p^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2p^4-4p^3+6p^2-4p+1+4\cdot(1-p)p^3+4(1-3p+3p^2-p^3)p+6(1-2p+p^2)p^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}2p^4-4p^3+6p^2-4p+1+(4-4p)p^3+(4-12p+12p^2-4p^3)p+(6-12p+6p^2)p^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}2p^4-4p^3+6p^2-4p+1+4p^3-4p^4+4p-12p^2+12p^3-4p^4+6p^2-12p^3+6p^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}-2p^4+6p^2-4p+1+4p-12p^2+12p^3-4p^4+6p^2-12p^3+6p^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}-6p^4+12p^3-6p^2+1+6p^2-12p^3+6p^4 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}1\end{aligned} $$ | |
| ① | $$ (1-p)^4 = (1-p)^2 \cdot (1-p)^2 $$ |
| ② | Find $ \left(1-p\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ p }$. $$ \begin{aligned}\left(1-p\right)^2 = \color{blue}{1^2} -2 \cdot 1 \cdot p + \color{red}{p^2} = 1-2p+p^2\end{aligned} $$ |
| ③ | Multiply each term of $ \left( \color{blue}{1-2p+p^2}\right) $ by each term in $ \left( 1-2p+p^2\right) $. $$ \left( \color{blue}{1-2p+p^2}\right) \cdot \left( 1-2p+p^2\right) = 1-2p+p^2-2p+4p^2-2p^3+p^2-2p^3+p^4 $$ |
| ④ | Combine like terms: $$ 1 \color{blue}{-2p} + \color{red}{p^2} \color{blue}{-2p} + \color{green}{4p^2} \color{orange}{-2p^3} + \color{green}{p^2} \color{orange}{-2p^3} +p^4 = p^4 \color{orange}{-4p^3} + \color{green}{6p^2} \color{blue}{-4p} +1 $$Find $ \left(1-p\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 1 $ and $ B = p $. $$ \left(1-p\right)^3 = 1^3-3 \cdot 1^2 \cdot p + 3 \cdot 1 \cdot p^2-p^3 = 1-3p+3p^2-p^3 $$Find $ \left(1-p\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 1 } $ and $ B = \color{red}{ p }$. $$ \begin{aligned}\left(1-p\right)^2 = \color{blue}{1^2} -2 \cdot 1 \cdot p + \color{red}{p^2} = 1-2p+p^2\end{aligned} $$ |
| ⑤ | Combine like terms: $$ \color{blue}{p^4} -4p^3+6p^2-4p+1+ \color{blue}{p^4} = \color{blue}{2p^4} -4p^3+6p^2-4p+1 $$ |
| ⑥ | Multiply $ \color{blue}{4} $ by $ \left( 1-p\right) $ $$ \color{blue}{4} \cdot \left( 1-p\right) = 4-4p $$Multiply $ \color{blue}{4} $ by $ \left( 1-3p+3p^2-p^3\right) $ $$ \color{blue}{4} \cdot \left( 1-3p+3p^2-p^3\right) = 4-12p+12p^2-4p^3 $$Multiply $ \color{blue}{6} $ by $ \left( 1-2p+p^2\right) $ $$ \color{blue}{6} \cdot \left( 1-2p+p^2\right) = 6-12p+6p^2 $$ |
| ⑦ | $$ \left( \color{blue}{4-4p}\right) \cdot p^3 = 4p^3-4p^4 $$$$ \left( \color{blue}{4-12p+12p^2-4p^3}\right) \cdot p = 4p-12p^2+12p^3-4p^4 $$$$ \left( \color{blue}{6-12p+6p^2}\right) \cdot p^2 = 6p^2-12p^3+6p^4 $$ |
| ⑧ | Combine like terms: $$ \color{blue}{2p^4} \, \color{red}{ -\cancel{4p^3}} \,+6p^2-4p+1+ \, \color{red}{ \cancel{4p^3}} \, \color{blue}{-4p^4} = \color{blue}{-2p^4} +6p^2-4p+1 $$ |
| ⑨ | Combine like terms: $$ \color{blue}{-2p^4} + \color{red}{6p^2} \, \color{green}{ -\cancel{4p}} \,+1+ \, \color{green}{ \cancel{4p}} \, \color{red}{-12p^2} +12p^3 \color{blue}{-4p^4} = \color{blue}{-6p^4} +12p^3 \color{red}{-6p^2} +1 $$ |
| ⑩ | Combine like terms: $$ \, \color{blue}{ -\cancel{6p^4}} \,+ \, \color{green}{ \cancel{12p^3}} \, \, \color{blue}{ -\cancel{6p^2}} \,+1+ \, \color{blue}{ \cancel{6p^2}} \, \, \color{green}{ -\cancel{12p^3}} \,+ \, \color{blue}{ \cancel{6p^4}} \, = 1 $$ |
This page was created using
Simplify polynomials calculator