Question
Answer
$$(x-1)^2(x-2)(x-3)(x-4)(x-5)=x^6-16x^5+100x^4-310x^3+499x^2-394x+120$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(x-1)^2(x-2)(x-3)(x-4)(x-5)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-2x+1)(x-2)(x-3)(x-4)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-2x^2-2x^2+4x+x-2)(x-3)(x-4)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-4x^2+5x-2)(x-3)(x-4)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^4-7x^3+17x^2-17x+6)(x-4)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(x^5-11x^4+45x^3-85x^2+74x-24)(x-5) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}x^6-16x^5+100x^4-310x^3+499x^2-394x+120\end{aligned} $$ | |
| ① | Find $ \left(x-1\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ x } $ and $ B = \color{red}{ 1 }$. $$ \begin{aligned}\left(x-1\right)^2 = \color{blue}{x^2} -2 \cdot x \cdot 1 + \color{red}{1^2} = x^2-2x+1\end{aligned} $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-2x+1}\right) $ by each term in $ \left( x-2\right) $. $$ \left( \color{blue}{x^2-2x+1}\right) \cdot \left( x-2\right) = x^3-2x^2-2x^2+4x+x-2 $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-2x^2} \color{blue}{-2x^2} + \color{red}{4x} + \color{red}{x} -2 = x^3 \color{blue}{-4x^2} + \color{red}{5x} -2 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-4x^2+5x-2}\right) $ by each term in $ \left( x-3\right) $. $$ \left( \color{blue}{x^3-4x^2+5x-2}\right) \cdot \left( x-3\right) = x^4-3x^3-4x^3+12x^2+5x^2-15x-2x+6 $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-3x^3} \color{blue}{-4x^3} + \color{red}{12x^2} + \color{red}{5x^2} \color{green}{-15x} \color{green}{-2x} +6 = x^4 \color{blue}{-7x^3} + \color{red}{17x^2} \color{green}{-17x} +6 $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^4-7x^3+17x^2-17x+6}\right) $ by each term in $ \left( x-4\right) $. $$ \left( \color{blue}{x^4-7x^3+17x^2-17x+6}\right) \cdot \left( x-4\right) = x^5-4x^4-7x^4+28x^3+17x^3-68x^2-17x^2+68x+6x-24 $$ |
| ⑦ | Combine like terms: $$ x^5 \color{blue}{-4x^4} \color{blue}{-7x^4} + \color{red}{28x^3} + \color{red}{17x^3} \color{green}{-68x^2} \color{green}{-17x^2} + \color{orange}{68x} + \color{orange}{6x} -24 = \\ = x^5 \color{blue}{-11x^4} + \color{red}{45x^3} \color{green}{-85x^2} + \color{orange}{74x} -24 $$ |
| ⑧ | Multiply each term of $ \left( \color{blue}{x^5-11x^4+45x^3-85x^2+74x-24}\right) $ by each term in $ \left( x-5\right) $. $$ \left( \color{blue}{x^5-11x^4+45x^3-85x^2+74x-24}\right) \cdot \left( x-5\right) = \\ = x^6-5x^5-11x^5+55x^4+45x^4-225x^3-85x^3+425x^2+74x^2-370x-24x+120 $$ |
| ⑨ | Combine like terms: $$ x^6 \color{blue}{-5x^5} \color{blue}{-11x^5} + \color{red}{55x^4} + \color{red}{45x^4} \color{green}{-225x^3} \color{green}{-85x^3} + \color{orange}{425x^2} + \color{orange}{74x^2} \color{blue}{-370x} \color{blue}{-24x} +120 = \\ = x^6 \color{blue}{-16x^5} + \color{red}{100x^4} \color{green}{-310x^3} + \color{orange}{499x^2} \color{blue}{-394x} +120 $$ |
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