Question
Answer
$$14.5\cdot\dfrac{(x-1970)(x-2010)(x-2020)}{(1990-1970)\cdot(1990-2010)\cdot(1990-2020)}=\dfrac{14x^3-84000x^2+14x-111980316000}{12000}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}14.5 \cdot \frac{(x-1970)(x-2010)(x-2020)}{(1990-1970)\cdot(1990-2010)\cdot(1990-2020)}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}14.5 \cdot \frac{(x^2-2010x-1970x+3959700)(x-2020)}{(3960100-3999900-3920300+3959700)\cdot(-30)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}14.5 \cdot \frac{(x^2-3980x+3959700)(x-2020)}{-400\cdot(-30)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}14.5 \cdot \frac{(x^2-3980x+3959700)(x-2020)}{12000} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}14.5 \cdot \frac{x^3-2020x^2-3980x^2+8039600x+3959700x-7998594000}{12000} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}14.5 \cdot \frac{x^3-6000x^2+11999300x-7998594000}{12000} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{14x^3-84000x^2+14x-111980316000}{12000}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{x-1970}\right) $ by each term in $ \left( x-2010\right) $. $$ \left( \color{blue}{x-1970}\right) \cdot \left( x-2010\right) = x^2-2010x-1970x+3959700 $$ |
| ② | Multiply each term of $ \left( \color{blue}{1990-1970}\right) $ by each term in $ \left( 1990-2010\right) $. $$ \left( \color{blue}{1990-1970}\right) \cdot \left( 1990-2010\right) = 3960100-3999900-3920300+3959700 $$ |
| ③ | Combine like terms: $$ x^2 \color{blue}{-2010x} \color{blue}{-1970x} +3959700 = x^2 \color{blue}{-3980x} +3959700 $$ |
| ④ | Combine like terms: $$ \color{blue}{3960100} \color{red}{-3999900} \color{green}{-3920300} + \color{green}{3959700} = \color{green}{-400} $$ |
| ⑤ | Combine like terms: $$ x^2 \color{blue}{-2010x} \color{blue}{-1970x} +3959700 = x^2 \color{blue}{-3980x} +3959700 $$ |
| ⑥ | $ -400 \cdot ( -30 ) = 12000 $ |
| ⑦ | Multiply each term of $ \left( \color{blue}{x^2-3980x+3959700}\right) $ by each term in $ \left( x-2020\right) $. $$ \left( \color{blue}{x^2-3980x+3959700}\right) \cdot \left( x-2020\right) = x^3-2020x^2-3980x^2+8039600x+3959700x-7998594000 $$ |
| ⑧ | Combine like terms: $$ x^3 \color{blue}{-2020x^2} \color{blue}{-3980x^2} + \color{red}{8039600x} + \color{red}{3959700x} -7998594000 = x^3 \color{blue}{-6000x^2} + \color{red}{11999300x} -7998594000 $$ |
| ⑨ | Multiply $14$ by $ \dfrac{x^3-6000x^2+x-7998594000}{12000} $ to get $ \dfrac{ 14x^3-84000x^2+14x-111980316000 }{ 12000 } $. Step 1: Write $ 14 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 14 \cdot \frac{x^3-6000x^2+x-7998594000}{12000} & \xlongequal{\text{Step 1}} \frac{14}{\color{red}{1}} \cdot \frac{x^3-6000x^2+x-7998594000}{12000} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 14 \cdot \left( x^3-6000x^2+x-7998594000 \right) }{ 1 \cdot 12000 } \xlongequal{\text{Step 3}} \frac{ 14x^3-84000x^2+14x-111980316000 }{ 12000 } \end{aligned} $$ |
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