Question
Answer
$$15\cdot\dfrac{(x-1990)(x-2010)(x-2020)}{(1970-1990)\cdot(1970-2010)\cdot(1970-2020)}=\dfrac{15x^3-90300x^2+15x-121196970000}{-40000}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}15 \cdot \frac{(x-1990)(x-2010)(x-2020)}{(1970-1990)\cdot(1970-2010)\cdot(1970-2020)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}15 \cdot \frac{(x^2-2010x-1990x+3999900)(x-2020)}{(3880900-3959700-3920300+3999900)\cdot(-50)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}15 \cdot \frac{(x^2-4000x+3999900)(x-2020)}{800\cdot(-50)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}15 \cdot \frac{(x^2-4000x+3999900)(x-2020)}{-40000} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}15 \cdot \frac{x^3-2020x^2-4000x^2+8080000x+3999900x-8079798000}{-40000} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}15 \cdot \frac{x^3-6020x^2+12079900x-8079798000}{-40000} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{15x^3-90300x^2+15x-121196970000}{-40000}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-1990}\right) $ by each term in $ \left( x-2010\right) $. $$ \left( \color{blue}{x-1990}\right) \cdot \left( x-2010\right) = x^2-2010x-1990x+3999900 $$ |
| ② | Multiply each term of $ \left( \color{blue}{1970-1990}\right) $ by each term in $ \left( 1970-2010\right) $. $$ \left( \color{blue}{1970-1990}\right) \cdot \left( 1970-2010\right) = 3880900-3959700-3920300+3999900 $$ |
| ③ | Combine like terms: $$ x^2 \color{blue}{-2010x} \color{blue}{-1990x} +3999900 = x^2 \color{blue}{-4000x} +3999900 $$ |
| ④ | Combine like terms: $$ \color{blue}{3880900} \color{red}{-3959700} \color{green}{-3920300} + \color{green}{3999900} = \color{green}{800} $$ |
| ⑤ | Combine like terms: $$ x^2 \color{blue}{-2010x} \color{blue}{-1990x} +3999900 = x^2 \color{blue}{-4000x} +3999900 $$ |
| ⑥ | $ 800 \cdot ( -50 ) = -40000 $ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^2-4000x+3999900}\right) $ by each term in $ \left( x-2020\right) $. $$ \left( \color{blue}{x^2-4000x+3999900}\right) \cdot \left( x-2020\right) = x^3-2020x^2-4000x^2+8080000x+3999900x-8079798000 $$ |
| ⑧ | Combine like terms: $$ x^3 \color{blue}{-2020x^2} \color{blue}{-4000x^2} + \color{red}{8080000x} + \color{red}{3999900x} -8079798000 = x^3 \color{blue}{-6020x^2} + \color{red}{12079900x} -8079798000 $$ |
| ⑨ | Multiply $15$ by $ \dfrac{x^3-6020x^2+x-8079798000}{-40000} $ to get $ \dfrac{ 15x^3-90300x^2+15x-121196970000 }{ -40000 } $. Step 1: Write $ 15 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 15 \cdot \frac{x^3-6020x^2+x-8079798000}{-40000} & \xlongequal{\text{Step 1}} \frac{15}{\color{red}{1}} \cdot \frac{x^3-6020x^2+x-8079798000}{-40000} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 15 \cdot \left( x^3-6020x^2+x-8079798000 \right) }{ 1 \cdot \left( -40000 \right) } \xlongequal{\text{Step 3}} \frac{ 15x^3-90300x^2+15x-121196970000 }{ -40000 } \end{aligned} $$ |
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