Question
Answer
$$x^3+6x^2+11x+30-(x-5)(x^4+x^3+4x^2-2x-12-(x-5)x^3)=-6x^4+27x^3+28x^2+13x-30$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x^3+6x^2+11x+30-(x-5)(x^4+x^3+4x^2-2x-12-(x-5)x^3)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}x^3+6x^2+11x+30-(x-5)(x^4+x^3+4x^2-2x-12-(x^4-5x^3)) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}x^3+6x^2+11x+30-(x-5)(x^4+x^3+4x^2-2x-12-x^4+5x^3) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}x^3+6x^2+11x+30-(x-5)(6x^3+4x^2-2x-12) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}x^3+6x^2+11x+30-(6x^4-26x^3-22x^2-2x+60) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}x^3+6x^2+11x+30-6x^4+26x^3+22x^2+2x-60 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}-6x^4+27x^3+28x^2+13x-30\end{aligned} $$ | |
| ① | $$ \left( \color{blue}{x-5}\right) \cdot x^3 = x^4-5x^3 $$ |
| ② | Remove the parentheses by changing the sign of each term within them. $$ - \left( x^4-5x^3 \right) = -x^4+5x^3 $$ |
| ③ | Combine like terms: $$ \, \color{blue}{ \cancel{x^4}} \,+ \color{green}{x^3} +4x^2-2x-12 \, \color{blue}{ -\cancel{x^4}} \,+ \color{green}{5x^3} = \color{green}{6x^3} +4x^2-2x-12 $$ |
| ④ | Multiply each term of $ \left( \color{blue}{x-5}\right) $ by each term in $ \left( 6x^3+4x^2-2x-12\right) $. $$ \left( \color{blue}{x-5}\right) \cdot \left( 6x^3+4x^2-2x-12\right) = 6x^4+4x^3-2x^2-12x-30x^3-20x^2+10x+60 $$ |
| ⑤ | Combine like terms: $$ 6x^4+ \color{blue}{4x^3} \color{red}{-2x^2} \color{green}{-12x} \color{blue}{-30x^3} \color{red}{-20x^2} + \color{green}{10x} +60 = 6x^4 \color{blue}{-26x^3} \color{red}{-22x^2} \color{green}{-2x} +60 $$ |
| ⑥ | Remove the parentheses by changing the sign of each term within them. $$ - \left( 6x^4-26x^3-22x^2-2x+60 \right) = -6x^4+26x^3+22x^2+2x-60 $$ |
| ⑦ | Combine like terms: $$ \color{blue}{x^3} + \color{red}{6x^2} + \color{green}{11x} + \color{orange}{30} -6x^4+ \color{blue}{26x^3} + \color{red}{22x^2} + \color{green}{2x} \color{orange}{-60} = \\ = -6x^4+ \color{blue}{27x^3} + \color{red}{28x^2} + \color{green}{13x} \color{orange}{-30} $$ |
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