Distributiviteit \((a+r)(b+s)\)

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Werk uit en vereenvoudig

1. \((a-\frac{-9}{9})(p+\frac{-9}{9})\)
2. \((p-\frac{9}{14})(x+(-20))\)
3. \((a+\frac{11}{6})(x-(-10))\)
4. \((q+\frac{19}{10})(p+\frac{-2}{8})\)
5. \((p+\frac{1}{13})(\frac{19}{4}-q)\)
6. \((\frac{-13}{20}-p)(p+\frac{14}{14})\)
7. \((\frac{-10}{4}+p)(\frac{-10}{10}+b)\)
8. \((a-20)(q+\frac{8}{10})\)
9. \((p+\frac{15}{4})((-17)+p)\)
10. \((q+\frac{17}{6})(\frac{2}{9}-q)\)
11. \((p+\frac{-19}{15})(\frac{-20}{2}-q)\)
12. \((\frac{20}{19}-a)(\frac{-4}{10}-a)\)

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Werk uit en vereenvoudig

Verbetersleutel

1. \(\begin{align}(a-\frac{-9}{9})(p+\frac{-9}{9})& =(\color{red}{a}+\color{orange}{1}).(\color{green}{p}+\color{blue}{(-1)})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(p)} + \color{red}{(a)} . \color{blue}{(-1)} + \color{orange}{(1)} . \color{green}{(p)} + \color{orange}{(1)} . \color{blue}{(-1)}& \text{Boogjes en uitschrijven} \\ & = ap-1a+1p-1.1& \text{Uitwerken } \\ & =ap-1a+1p-1 & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
2. \(\begin{align}(p-\frac{9}{14})(x+(-20))& =(\color{red}{p}+\color{orange}{\frac{-9}{14}}).(\color{green}{x}+\color{blue}{(-20)})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(x)} + \color{red}{(p)} . \color{blue}{(-20)} + \color{orange}{(\frac{-9}{14})} . \color{green}{(x)} + \color{orange}{(\frac{-9}{14})} . \color{blue}{(-20)}& \text{Boogjes en uitschrijven} \\ & = px-20p-\frac{9}{14}x+\frac{9}{14}.20& \text{Uitwerken } \\ & =px-20p-\frac{9}{14}x+\frac{180}{14} & \text{Uitrekenen (Geen Smoothie!)} \\ & = px-20p-\frac{9}{14}x+\frac{90}{7} & \text{Vereenvoudig} \\\end{align}\)
3. \(\begin{align}(a+\frac{11}{6})(x-(-10))& =(\color{red}{a}+\color{orange}{\frac{11}{6}}).(\color{green}{x}+\color{blue}{10})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(x)} + \color{red}{(a)} . \color{blue}{(10)} + \color{orange}{(\frac{11}{6})} . \color{green}{(x)} + \color{orange}{(\frac{11}{6})} . \color{blue}{(10)}& \text{Boogjes en uitschrijven} \\ & = ax+10a+\frac{11}{6}x+\frac{11}{6}.10& \text{Uitwerken } \\ & =ax+10a+\frac{11}{6}x+\frac{110}{6} & \text{Uitrekenen (Geen Smoothie!)} \\ & = ax+10a+\frac{11}{6}x+\frac{55}{3} & \text{Vereenvoudig} \\\end{align}\)
4. \(\begin{align}(q+\frac{19}{10})(p+\frac{-2}{8})& =(\color{red}{q}+\color{orange}{\frac{19}{10}}).(\color{green}{p}+\color{blue}{\frac{-1}{4}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(p)} + \color{red}{(q)} . \color{blue}{(\frac{-1}{4})} + \color{orange}{(\frac{19}{10})} . \color{green}{(p)} + \color{orange}{(\frac{19}{10})} . \color{blue}{(\frac{-1}{4})}& \text{Boogjes en uitschrijven} \\ & = pq-\frac{1}{4}q+\frac{19}{10}p-\frac{19}{10}.\frac{1}{4}& \text{Uitwerken } \\ & =pq-\frac{1}{4}q+\frac{19}{10}p-\frac{19}{40} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
5. \(\begin{align}(p+\frac{1}{13})(\frac{19}{4}-q)& =(\color{red}{p}+\color{orange}{\frac{1}{13}}).(\color{green}{-q}+\color{blue}{\frac{19}{4}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(-q)} + \color{red}{(p)} . \color{blue}{(\frac{19}{4})} + \color{orange}{(\frac{1}{13})} . \color{green}{(-q)} + \color{orange}{(\frac{1}{13})} . \color{blue}{(\frac{19}{4})}& \text{Boogjes en uitschrijven} \\ & = -pq+\frac{19}{4}p-\frac{1}{13}q+\frac{1}{13}.\frac{19}{4}& \text{Uitwerken } \\ & =-pq+\frac{19}{4}p-\frac{1}{13}q+\frac{19}{52} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
6. \(\begin{align}(\frac{-13}{20}-p)(p+\frac{14}{14})& =(\color{red}{-p}+\color{orange}{\frac{-13}{20}}).(\color{green}{p}+\color{blue}{1})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-p)} . \color{green}{(p)} + \color{red}{(-p)} . \color{blue}{(1)} + \color{orange}{(\frac{-13}{20})} . \color{green}{(p)} + \color{orange}{(\frac{-13}{20})} . \color{blue}{(1)}& \text{Boogjes en uitschrijven} \\ & = -pp-1p-\frac{13}{20}p-\frac{13}{20}.1& \text{Uitwerken } \\ & =-p^2+((-1)+(\frac{-13}{20}))p-\frac{13}{20} & \text{Neem samen} \\ &=-p^2-\frac{33}{20}p-\frac{13}{20} & \text{Maak de som} \\\end{align}\)
7. \(\begin{align}(\frac{-10}{4}+p)(\frac{-10}{10}+b)& =(\color{red}{p}+\color{orange}{\frac{-5}{2}}).(\color{green}{b}+\color{blue}{(-1)})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(b)} + \color{red}{(p)} . \color{blue}{(-1)} + \color{orange}{(\frac{-5}{2})} . \color{green}{(b)} + \color{orange}{(\frac{-5}{2})} . \color{blue}{(-1)}& \text{Boogjes en uitschrijven} \\ & = bp-1p-\frac{5}{2}b+\frac{5}{2}.1& \text{Uitwerken } \\ & =bp-1p-\frac{5}{2}b+\frac{5}{2} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
8. \(\begin{align}(a-20)(q+\frac{8}{10})& =(\color{red}{a}+\color{orange}{(-20)}).(\color{green}{q}+\color{blue}{\frac{4}{5}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(q)} + \color{red}{(a)} . \color{blue}{(\frac{4}{5})} + \color{orange}{(-20)} . \color{green}{(q)} + \color{orange}{(-20)} . \color{blue}{(\frac{4}{5})}& \text{Boogjes en uitschrijven} \\ & = aq+\frac{4}{5}a-20q-20.\frac{4}{5}& \text{Uitwerken } \\ & =aq+\frac{4}{5}a-20q-\frac{80}{5} & \text{Uitrekenen (Geen Smoothie!)} \\ & = aq+\frac{4}{5}a-20q-16 & \text{Vereenvoudig} \\\end{align}\)
9. \(\begin{align}(p+\frac{15}{4})((-17)+p)& =(\color{red}{p}+\color{orange}{\frac{15}{4}}).(\color{green}{p}+\color{blue}{(-17)})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(p)} + \color{red}{(p)} . \color{blue}{(-17)} + \color{orange}{(\frac{15}{4})} . \color{green}{(p)} + \color{orange}{(\frac{15}{4})} . \color{blue}{(-17)}& \text{Boogjes en uitschrijven} \\ & = pp-17p+\frac{15}{4}p-\frac{15}{4}.17& \text{Uitwerken } \\ & =p^2+((-17)+\frac{15}{4})p-(\frac{-255}{4}) & \text{Neem samen} \\ &=p^2-\frac{53}{4}p-(\frac{-255}{4}) & \text{Maak de som} \\\end{align}\)
10. \(\begin{align}(q+\frac{17}{6})(\frac{2}{9}-q)& =(\color{red}{q}+\color{orange}{\frac{17}{6}}).(\color{green}{-q}+\color{blue}{\frac{2}{9}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(-q)} + \color{red}{(q)} . \color{blue}{(\frac{2}{9})} + \color{orange}{(\frac{17}{6})} . \color{green}{(-q)} + \color{orange}{(\frac{17}{6})} . \color{blue}{(\frac{2}{9})}& \text{Boogjes en uitschrijven} \\ & = -qq+\frac{2}{9}q-\frac{17}{6}q+\frac{17}{6}.\frac{2}{9}& \text{Uitwerken } \\ & =-q^2+(\frac{2}{9}+(\frac{-17}{6}))q+(\frac{-34}{54}) & \text{Neem samen} \\ &=-q^2-\frac{47}{18}q+(\frac{-34}{54}) & \text{Maak de som} \\ &=-q^2-\frac{47}{18}q+(\frac{-17}{27}) & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)
11. \(\begin{align}(p+\frac{-19}{15})(\frac{-20}{2}-q)& =(\color{red}{p}+\color{orange}{\frac{-19}{15}}).(\color{green}{-q}+\color{blue}{(-10)})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(-q)} + \color{red}{(p)} . \color{blue}{(-10)} + \color{orange}{(\frac{-19}{15})} . \color{green}{(-q)} + \color{orange}{(\frac{-19}{15})} . \color{blue}{(-10)}& \text{Boogjes en uitschrijven} \\ & = -pq-10p+\frac{19}{15}q+\frac{19}{15}.10& \text{Uitwerken } \\ & =-pq-10p+\frac{19}{15}q+\frac{190}{15} & \text{Uitrekenen (Geen Smoothie!)} \\ & = -pq-10p+\frac{19}{15}q+\frac{38}{3} & \text{Vereenvoudig} \\\end{align}\)
12. \(\begin{align}(\frac{20}{19}-a)(\frac{-4}{10}-a)& =(\color{red}{-a}+\color{orange}{\frac{20}{19}}).(\color{green}{-a}+\color{blue}{\frac{-2}{5}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-a)} . \color{green}{(-a)} + \color{red}{(-a)} . \color{blue}{(\frac{-2}{5})} + \color{orange}{(\frac{20}{19})} . \color{green}{(-a)} + \color{orange}{(\frac{20}{19})} . \color{blue}{(\frac{-2}{5})}& \text{Boogjes en uitschrijven} \\ & = aa+\frac{2}{5}a-\frac{20}{19}a-\frac{20}{19}.\frac{2}{5}& \text{Uitwerken } \\ & =a^2+(\frac{2}{5}+(\frac{-20}{19}))a-(\frac{-40}{95}) & \text{Neem samen} \\ &=a^2-\frac{62}{95}a-(\frac{-40}{95}) & \text{Maak de som} \\ &=a^2-\frac{62}{95}a+\frac{8}{19} & \text{Vereenvoudig indien mogelijk 1} \\\end{align}\)