Question
Answer
$$(x-2)(x-3)^3(x^2-1)=x^6-11x^5+44x^4-70x^3+9x^2+81x-54$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-2)(x-3)^3(x^2-1)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x-2)(x^3-9x^2+27x-27)(x^2-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^4-11x^3+45x^2-81x+54)(x^2-1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^6-11x^5+44x^4-70x^3+9x^2+81x-54\end{aligned} $$ | |
| ① | Find $ \left(x-3\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = x $ and $ B = 3 $. $$ \left(x-3\right)^3 = x^3-3 \cdot x^2 \cdot 3 + 3 \cdot x \cdot 3^2-3^3 = x^3-9x^2+27x-27 $$ |
| ② | Multiply each term of $ \left( \color{blue}{x-2}\right) $ by each term in $ \left( x^3-9x^2+27x-27\right) $. $$ \left( \color{blue}{x-2}\right) \cdot \left( x^3-9x^2+27x-27\right) = x^4-9x^3+27x^2-27x-2x^3+18x^2-54x+54 $$ |
| ③ | Combine like terms: $$ x^4 \color{blue}{-9x^3} + \color{red}{27x^2} \color{green}{-27x} \color{blue}{-2x^3} + \color{red}{18x^2} \color{green}{-54x} +54 = x^4 \color{blue}{-11x^3} + \color{red}{45x^2} \color{green}{-81x} +54 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^4-11x^3+45x^2-81x+54}\right) $ by each term in $ \left( x^2-1\right) $. $$ \left( \color{blue}{x^4-11x^3+45x^2-81x+54}\right) \cdot \left( x^2-1\right) = x^6-x^4-11x^5+11x^3+45x^4-45x^2-81x^3+81x+54x^2-54 $$ |
| ⑤ | Combine like terms: $$ x^6 \color{blue}{-x^4} -11x^5+ \color{red}{11x^3} + \color{blue}{45x^4} \color{green}{-45x^2} \color{red}{-81x^3} +81x+ \color{green}{54x^2} -54 = \\ = x^6-11x^5+ \color{blue}{44x^4} \color{red}{-70x^3} + \color{green}{9x^2} +81x-54 $$ |
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