Question
Answer
$$(x-5)(x+5)(x-1)(x+1)=x^4-26x^2+25$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-5)(x+5)(x-1)(x+1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2+5x-5x-25)(x-1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^2-25)(x-1)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-x^2-25x+25)(x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^4-26x^2+25\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-5}\right) $ by each term in $ \left( x+5\right) $. $$ \left( \color{blue}{x-5}\right) \cdot \left( x+5\right) = x^2+ \cancel{5x} -\cancel{5x}-25 $$ |
| ② | Combine like terms: $$ x^2+ \, \color{blue}{ \cancel{5x}} \, \, \color{blue}{ -\cancel{5x}} \,-25 = x^2-25 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^2-25}\right) $ by each term in $ \left( x-1\right) $. $$ \left( \color{blue}{x^2-25}\right) \cdot \left( x-1\right) = x^3-x^2-25x+25 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-x^2-25x+25}\right) $ by each term in $ \left( x+1\right) $. $$ \left( \color{blue}{x^3-x^2-25x+25}\right) \cdot \left( x+1\right) = \\ = x^4+ \cancel{x^3} -\cancel{x^3}-x^2-25x^2 -\cancel{25x}+ \cancel{25x}+25 $$ |
| ⑤ | Combine like terms: $$ x^4+ \, \color{blue}{ \cancel{x^3}} \, \, \color{blue}{ -\cancel{x^3}} \, \color{green}{-x^2} \color{green}{-25x^2} \, \color{orange}{ -\cancel{25x}} \,+ \, \color{orange}{ \cancel{25x}} \,+25 = x^4 \color{green}{-26x^2} +25 $$ |
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