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Question

Answer

$$\dfrac{m-1}{2}m^2-21m+19=\dfrac{m^3-m^2-42m+38}{2}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{m-1}{2}m^2-21m+19& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{m^3-m^2}{2}-21m+19 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{m^3-m^2-42m}{2}+19 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{m^3-m^2-42m+38}{2}\end{aligned} $$
①

Multiply $ \dfrac{m-1}{2} $ by $ m^2 $ to get $ \dfrac{ m^3-m^2 }{ 2 } $.

Step 1: Write $ m^2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{m-1}{2} \cdot m^2 & \xlongequal{\text{Step 1}} \frac{m-1}{2} \cdot \frac{m^2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( m-1 \right) \cdot m^2 }{ 2 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ m^3-m^2 }{ 2 } \end{aligned} $$
②

Subtract $21m$ from $ \dfrac{m^3-m^2}{2} $ to get $ \dfrac{ \color{purple}{ m^3-m^2-42m } }{ 2 }$.

Step 1: Write $ 21m $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{m^3-m^2}{2} -21m & \xlongequal{\text{Step 1}} \frac{m^3-m^2}{2} - \frac{21m}{\color{red}{1}} = \frac{ m^3-m^2 }{ 2 } - \frac{ 21m \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ m^3-m^2 } }{ 2 } - \frac{ \color{purple}{ 42m } }{ 2 }=\frac{ \color{purple}{ m^3-m^2-42m } }{ 2 } \end{aligned} $$
③

Add $ \dfrac{m^3-m^2-42m}{2} $ and $ 19 $ to get $ \dfrac{ \color{purple}{ m^3-m^2-42m+38 } }{ 2 }$.

Step 1: Write $ 19 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{m^3-m^2-42m}{2} +19 & \xlongequal{\text{Step 1}} \frac{m^3-m^2-42m}{2} + \frac{19}{\color{red}{1}} = \frac{ m^3-m^2-42m }{ 2 } + \frac{ 19 \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ m^3-m^2-42m } }{ 2 } + \frac{ \color{purple}{ 38 } }{ 2 }=\frac{ \color{purple}{ m^3-m^2-42m+38 } }{ 2 } \end{aligned} $$
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