Question
Answer
$$((2+sqrt\cdot7)x-1)((2-sqrt\cdot7)x-1)(x-1)=-49q^2r^2s^2t^2x^3+49q^2r^2s^2t^2x^2+4x^3-8x^2+5x-1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}((2+sqrt\cdot7)x-1)((2-sqrt\cdot7)x-1)(x-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(2x+7qrstx-1)(2x-7qrstx-1)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(-49q^2r^2s^2t^2x^2+4x^2-4x+1)(x-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-49q^2r^2s^2t^2x^3+49q^2r^2s^2t^2x^2+4x^3-8x^2+5x-1\end{aligned} $$ | |
| ① | $$ \left( \color{blue}{2+7qrst}\right) \cdot x = 2x+7qrstx $$$$ \left( \color{blue}{2-7qrst}\right) \cdot x = 2x-7qrstx $$ |
| ② | Multiply each term of $ \left( \color{blue}{2x+7qrstx-1}\right) $ by each term in $ \left( 2x-7qrstx-1\right) $. $$ \left( \color{blue}{2x+7qrstx-1}\right) \cdot \left( 2x-7qrstx-1\right) = \\ = 4x^2 -\cancel{14qrstx^2}-2x+ \cancel{14qrstx^2}-49q^2r^2s^2t^2x^2 -\cancel{7qrstx}-2x+ \cancel{7qrstx}+1 $$ |
| ③ | Combine like terms: $$ 4x^2 \, \color{blue}{ -\cancel{14qrstx^2}} \, \color{green}{-2x} + \, \color{blue}{ \cancel{14qrstx^2}} \,-49q^2r^2s^2t^2x^2 \, \color{orange}{ -\cancel{7qrstx}} \, \color{green}{-2x} + \, \color{orange}{ \cancel{7qrstx}} \,+1 = -49q^2r^2s^2t^2x^2+4x^2 \color{green}{-4x} +1 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{-49q^2r^2s^2t^2x^2+4x^2-4x+1}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{-49q^2r^2s^2t^2x^2+4x^2-4x+1}\right) \cdot \left( x-1\right) = \\ = -49q^2r^2s^2t^2x^3+49q^2r^2s^2t^2x^2+4x^3-4x^2-4x^2+4x+x-1 $$ |
| ⑤ | Combine like terms: $$ -49q^2r^2s^2t^2x^3+49q^2r^2s^2t^2x^2+4x^3 \color{blue}{-4x^2} \color{blue}{-4x^2} + \color{red}{4x} + \color{red}{x} -1 = \\ = -49q^2r^2s^2t^2x^3+49q^2r^2s^2t^2x^2+4x^3 \color{blue}{-8x^2} + \color{red}{5x} -1 $$ |
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