Question
Answer
$$(x^2-3)(x^2-5)(x^2-15)=x^6-23x^4+135x^2-225$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x^2-3)(x^2-5)(x^2-15)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^4-5x^2-3x^2+15)(x^2-15) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^4-8x^2+15)(x^2-15) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^6-15x^4-8x^4+120x^2+15x^2-225 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}x^6-23x^4+135x^2-225\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x^2-3}\right) $ by each term in $ \left( x^2-5\right) $. $$ \left( \color{blue}{x^2-3}\right) \cdot \left( x^2-5\right) = x^4-5x^2-3x^2+15 $$ |
| ② | Combine like terms: $$ x^4 \color{blue}{-5x^2} \color{blue}{-3x^2} +15 = x^4 \color{blue}{-8x^2} +15 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{x^4-8x^2+15}\right) $ by each term in $ \left( x^2-15\right) $. $$ \left( \color{blue}{x^4-8x^2+15}\right) \cdot \left( x^2-15\right) = x^6-15x^4-8x^4+120x^2+15x^2-225 $$ |
| ④ | Combine like terms: $$ x^6 \color{blue}{-15x^4} \color{blue}{-8x^4} + \color{red}{120x^2} + \color{red}{15x^2} -225 = x^6 \color{blue}{-23x^4} + \color{red}{135x^2} -225 $$ |
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