$$-\dfrac{14}{9}z(z-\dfrac{1}{4})(z-1)-\dfrac{1}{6}z(z-1)+\dfrac{14}{9}z(z-\dfrac{1}{4})^2=\dfrac{56z^3-76z^2+20z}{36}+\dfrac{14z}{9}(z-\dfrac{1}{4})^2$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-\frac{14}{9}z(z-\frac{1}{4})(z-1)-\frac{1}{6}z(z-1)+\frac{14}{9}z(z-\frac{1}{4})^2& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(-\frac{14}{9}z(z-\frac{1}{4})-\frac{1}{6}z)(z-1)+\frac{14z}{9}(z-\frac{1}{4})^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(\frac{14z}{9}\frac{4z-1}{4}-\frac{z}{6})(z-1)+\frac{14z}{9}(z-\frac{1}{4})^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}(\frac{56z^2-14z}{36}-\frac{z}{6})(z-1)+\frac{14z}{9}(z-\frac{1}{4})^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{56z^2-20z}{36}(z-1)+\frac{14z}{9}(z-\frac{1}{4})^2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle12}{\textcircled {12}} \htmlClass{explanationCircle explanationCircle13}{\textcircled {13}} } }}}\frac{56z^3-76z^2+20z}{36}+\frac{14z}{9}(z-\frac{1}{4})^2\end{aligned} $$ | |
| ① | Use the distributive property. |
| ② | Multiply $ \dfrac{14}{9} $ by $ z $ to get $ \dfrac{ 14z }{ 9 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{14}{9} \cdot z & \xlongequal{\text{Step 1}} \frac{14}{9} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 14 \cdot z }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14z }{ 9 } \end{aligned} $$ |
| ③ | Multiply $ \dfrac{14}{9} $ by $ z $ to get $ \dfrac{ 14z }{ 9 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{14}{9} \cdot z & \xlongequal{\text{Step 1}} \frac{14}{9} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 14 \cdot z }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14z }{ 9 } \end{aligned} $$ |
| ④ | Subtract $ \dfrac{1}{4} $ from $ z $ to get $ \dfrac{ \color{purple}{ 4z-1 } }{ 4 }$. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. |
| ⑤ | Multiply $ \dfrac{1}{6} $ by $ z $ to get $ \dfrac{ z }{ 6 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{6} \cdot z & \xlongequal{\text{Step 1}} \frac{1}{6} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot z }{ 6 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ z }{ 6 } \end{aligned} $$ |
| ⑥ | Multiply $ \dfrac{14}{9} $ by $ z $ to get $ \dfrac{ 14z }{ 9 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{14}{9} \cdot z & \xlongequal{\text{Step 1}} \frac{14}{9} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 14 \cdot z }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14z }{ 9 } \end{aligned} $$ |
| ⑦ | Multiply $ \dfrac{14z}{9} $ by $ \dfrac{4z-1}{4} $ to get $ \dfrac{ 56z^2-14z }{ 36 } $. Step 1: Multiply numerators and denominators. Step 2: Simplify numerator and denominator. $$ \begin{aligned} \frac{14z}{9} \cdot \frac{4z-1}{4} & \xlongequal{\text{Step 1}} \frac{ 14z \cdot \left( 4z-1 \right) }{ 9 \cdot 4 } \xlongequal{\text{Step 2}} \frac{ 56z^2-14z }{ 36 } \end{aligned} $$ |
| ⑧ | Multiply $ \dfrac{1}{6} $ by $ z $ to get $ \dfrac{ z }{ 6 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{6} \cdot z & \xlongequal{\text{Step 1}} \frac{1}{6} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot z }{ 6 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ z }{ 6 } \end{aligned} $$ |
| ⑨ | Multiply $ \dfrac{14}{9} $ by $ z $ to get $ \dfrac{ 14z }{ 9 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{14}{9} \cdot z & \xlongequal{\text{Step 1}} \frac{14}{9} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 14 \cdot z }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14z }{ 9 } \end{aligned} $$ |
| ⑩ | Subtract $ \dfrac{z}{6} $ from $ \dfrac{56z^2-14z}{36} $ to get $ \dfrac{ \color{purple}{ 56z^2-20z } }{ 36 }$. To subtract raitonal expressions, both fractions must have the same denominator. |
| ⑪ | Multiply $ \dfrac{14}{9} $ by $ z $ to get $ \dfrac{ 14z }{ 9 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{14}{9} \cdot z & \xlongequal{\text{Step 1}} \frac{14}{9} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 14 \cdot z }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14z }{ 9 } \end{aligned} $$ |
| ⑫ | Multiply $ \dfrac{56z^2-20z}{36} $ by $ z-1 $ to get $ \dfrac{56z^3-76z^2+20z}{36} $. Step 1: Write $ z-1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{56z^2-20z}{36} \cdot z-1 & \xlongequal{\text{Step 1}} \frac{56z^2-20z}{36} \cdot \frac{z-1}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( 56z^2-20z \right) \cdot \left( z-1 \right) }{ 36 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 56z^3-56z^2-20z^2+20z }{ 36 } = \frac{56z^3-76z^2+20z}{36} \end{aligned} $$ |
| ⑬ | Multiply $ \dfrac{14}{9} $ by $ z $ to get $ \dfrac{ 14z }{ 9 } $. Step 1: Write $ z $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{14}{9} \cdot z & \xlongequal{\text{Step 1}} \frac{14}{9} \cdot \frac{z}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 14 \cdot z }{ 9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 14z }{ 9 } \end{aligned} $$ |