Question
Answer
$$5(5x-6)^3-(x^2+2x-1)^2=-x^4+621x^3-2252x^2+2704x-1081$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}5(5x-6)^3-(x^2+2x-1)^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}5(125x^3-450x^2+540x-216)-(x^4+4x^3+2x^2-4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}625x^3-2250x^2+2700x-1080-(x^4+4x^3+2x^2-4x+1) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}625x^3-2250x^2+2700x-1080-x^4-4x^3-2x^2+4x-1 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-x^4+621x^3-2252x^2+2704x-1081\end{aligned} $$ | |
| ① | Find $ \left(5x-6\right)^3 $ using formula $$ (A - B) = A^3 - 3A^2B + 3AB^2 - B^3 $$where $ A = 5x $ and $ B = 6 $. $$ \left(5x-6\right)^3 = \left( 5x \right)^3-3 \cdot \left( 5x \right)^2 \cdot 6 + 3 \cdot 5x \cdot 6^2-6^3 = 125x^3-450x^2+540x-216 $$Multiply each term of $ \left( \color{blue}{x^2+2x-1}\right) $ by each term in $ \left( x^2+2x-1\right) $. $$ \left( \color{blue}{x^2+2x-1}\right) \cdot \left( x^2+2x-1\right) = x^4+2x^3-x^2+2x^3+4x^2-2x-x^2-2x+1 $$ |
| ② | Combine like terms: $$ x^4+ \color{blue}{2x^3} \color{red}{-x^2} + \color{blue}{2x^3} + \color{green}{4x^2} \color{orange}{-2x} \color{green}{-x^2} \color{orange}{-2x} +1 = x^4+ \color{blue}{4x^3} + \color{green}{2x^2} \color{orange}{-4x} +1 $$ |
| ③ | Multiply $ \color{blue}{5} $ by $ \left( 125x^3-450x^2+540x-216\right) $ $$ \color{blue}{5} \cdot \left( 125x^3-450x^2+540x-216\right) = 625x^3-2250x^2+2700x-1080 $$ |
| ④ | Remove the parentheses by changing the sign of each term within them. $$ - \left( x^4+4x^3+2x^2-4x+1 \right) = -x^4-4x^3-2x^2+4x-1 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{625x^3} \color{red}{-2250x^2} + \color{green}{2700x} \color{orange}{-1080} -x^4 \color{blue}{-4x^3} \color{red}{-2x^2} + \color{green}{4x} \color{orange}{-1} = \\ = -x^4+ \color{blue}{621x^3} \color{red}{-2252x^2} + \color{green}{2704x} \color{orange}{-1081} $$ |
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