Answer
$$x(x-7.75)(x-15.5)(x-23.25)(x-31)=x^5-76x^4+2006x^3-21356x^2+74865x$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x-7.75)(x-15.5)(x-23.25)(x-31)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-7x)(x-15.5)(x-23.25)(x-31) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(x^3-15x^2-7x^2+105x)(x-23.25)(x-31) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}(x^3-22x^2+105x)(x-23.25)(x-31) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(x^4-23x^3-22x^3+506x^2+105x^2-2415x)(x-31) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}(x^4-45x^3+611x^2-2415x)(x-31) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}x^5-76x^4+2006x^3-21356x^2+74865x\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( x-7\right) $ $$ \color{blue}{x} \cdot \left( x-7\right) = x^2-7x $$ |
| ② | Multiply each term of $ \left( \color{blue}{x^2-7x}\right) $ by each term in $ \left( x-15\right) $. $$ \left( \color{blue}{x^2-7x}\right) \cdot \left( x-15\right) = x^3-15x^2-7x^2+105x $$ |
| ③ | Combine like terms: $$ x^3 \color{blue}{-15x^2} \color{blue}{-7x^2} +105x = x^3 \color{blue}{-22x^2} +105x $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x^3-22x^2+105x}\right) $ by each term in $ \left( x-23\right) $. $$ \left( \color{blue}{x^3-22x^2+105x}\right) \cdot \left( x-23\right) = x^4-23x^3-22x^3+506x^2+105x^2-2415x $$ |
| ⑤ | Combine like terms: $$ x^4 \color{blue}{-23x^3} \color{blue}{-22x^3} + \color{red}{506x^2} + \color{red}{105x^2} -2415x = x^4 \color{blue}{-45x^3} + \color{red}{611x^2} -2415x $$ |
| ⑥ | Multiply each term of $ \left( \color{blue}{x^4-45x^3+611x^2-2415x}\right) $ by each term in $ \left( x-31\right) $. $$ \left( \color{blue}{x^4-45x^3+611x^2-2415x}\right) \cdot \left( x-31\right) = \\ = x^5-31x^4-45x^4+1395x^3+611x^3-18941x^2-2415x^2+74865x $$ |
| ⑦ | Combine like terms: $$ x^5 \color{blue}{-31x^4} \color{blue}{-45x^4} + \color{red}{1395x^3} + \color{red}{611x^3} \color{green}{-18941x^2} \color{green}{-2415x^2} +74865x = \\ = x^5 \color{blue}{-76x^4} + \color{red}{2006x^3} \color{green}{-21356x^2} +74865x $$ |
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