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

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Werk uit en vereenvoudig

1. \(((-15)-p)(b-\frac{8}{4})\)
2. \((\frac{18}{4}+x)(\frac{-12}{2}+q)\)
3. \((b+\frac{-3}{13})(\frac{13}{17}-b)\)
4. \((\frac{14}{17}-b)(x+\frac{14}{4})\)
5. \((\frac{-5}{9}-a)(\frac{3}{15}-a)\)
6. \((\frac{-9}{6}+x)(b-\frac{-3}{6})\)
7. \((p-\frac{18}{20})(\frac{-5}{19}+q)\)
8. \((\frac{20}{13}+x)(\frac{-11}{10}+x)\)
9. \((p-\frac{-4}{15})(\frac{19}{3}+x)\)
10. \((\frac{-6}{17}+a)(\frac{9}{12}+q)\)
11. \(((-6)+x)(q+\frac{-6}{8})\)
12. \((a+\frac{-1}{15})(10+q)\)

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Werk uit en vereenvoudig

Verbetersleutel

1. \(\begin{align}((-15)-p)(b-\frac{8}{4})& =(\color{red}{-p}+\color{orange}{(-15)}).(\color{green}{b}+\color{blue}{(-2)})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-p)} . \color{green}{(b)} + \color{red}{(-p)} . \color{blue}{(-2)} + \color{orange}{(-15)} . \color{green}{(b)} + \color{orange}{(-15)} . \color{blue}{(-2)}& \text{Boogjes en uitschrijven} \\ & = -bp+2p-15b+15.2& \text{Uitwerken } \\ & =-bp+2p-15b+30 & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
2. \(\begin{align}(\frac{18}{4}+x)(\frac{-12}{2}+q)& =(\color{red}{x}+\color{orange}{\frac{9}{2}}).(\color{green}{q}+\color{blue}{(-6)})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(q)} + \color{red}{(x)} . \color{blue}{(-6)} + \color{orange}{(\frac{9}{2})} . \color{green}{(q)} + \color{orange}{(\frac{9}{2})} . \color{blue}{(-6)}& \text{Boogjes en uitschrijven} \\ & = qx-6x+\frac{9}{2}q-\frac{9}{2}.6& \text{Uitwerken } \\ & =qx-6x+\frac{9}{2}q-\frac{54}{2} & \text{Uitrekenen (Geen Smoothie!)} \\ & = qx-6x+\frac{9}{2}q-27 & \text{Vereenvoudig} \\\end{align}\)
3. \(\begin{align}(b+\frac{-3}{13})(\frac{13}{17}-b)& =(\color{red}{b}+\color{orange}{\frac{-3}{13}}).(\color{green}{-b}+\color{blue}{\frac{13}{17}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(-b)} + \color{red}{(b)} . \color{blue}{(\frac{13}{17})} + \color{orange}{(\frac{-3}{13})} . \color{green}{(-b)} + \color{orange}{(\frac{-3}{13})} . \color{blue}{(\frac{13}{17})}& \text{Boogjes en uitschrijven} \\ & = -bb+\frac{13}{17}b+\frac{3}{13}b-\frac{3}{13}.\frac{13}{17}& \text{Uitwerken } \\ & =-b^2+(\frac{13}{17}+\frac{3}{13})b-\frac{39}{221} & \text{Neem samen} \\ &=-b^2+\frac{220}{221}b-\frac{39}{221} & \text{Maak de som} \\ &=-b^2+\frac{220}{221}b-\frac{3}{17} & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)
4. \(\begin{align}(\frac{14}{17}-b)(x+\frac{14}{4})& =(\color{red}{-b}+\color{orange}{\frac{14}{17}}).(\color{green}{x}+\color{blue}{\frac{7}{2}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-b)} . \color{green}{(x)} + \color{red}{(-b)} . \color{blue}{(\frac{7}{2})} + \color{orange}{(\frac{14}{17})} . \color{green}{(x)} + \color{orange}{(\frac{14}{17})} . \color{blue}{(\frac{7}{2})}& \text{Boogjes en uitschrijven} \\ & = -bx-\frac{7}{2}b+\frac{14}{17}x+\frac{14}{17}.\frac{7}{2}& \text{Uitwerken } \\ & =-bx-\frac{7}{2}b+\frac{14}{17}x+\frac{98}{34} & \text{Uitrekenen (Geen Smoothie!)} \\ & = -bx-\frac{7}{2}b+\frac{14}{17}x+\frac{49}{17} & \text{Vereenvoudig} \\\end{align}\)
5. \(\begin{align}(\frac{-5}{9}-a)(\frac{3}{15}-a)& =(\color{red}{-a}+\color{orange}{\frac{-5}{9}}).(\color{green}{-a}+\color{blue}{\frac{1}{5}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(-a)} . \color{green}{(-a)} + \color{red}{(-a)} . \color{blue}{(\frac{1}{5})} + \color{orange}{(\frac{-5}{9})} . \color{green}{(-a)} + \color{orange}{(\frac{-5}{9})} . \color{blue}{(\frac{1}{5})}& \text{Boogjes en uitschrijven} \\ & = aa-\frac{1}{5}a+\frac{5}{9}a-\frac{5}{9}.\frac{1}{5}& \text{Uitwerken } \\ & =a^2+((\frac{-1}{5})+\frac{5}{9})a-(\frac{-5}{45}) & \text{Neem samen} \\ &=a^2+\frac{16}{45}a-(\frac{-5}{45}) & \text{Maak de som} \\ &=a^2+\frac{16}{45}a+\frac{1}{9} & \text{Vereenvoudig indien mogelijk 1} \\\end{align}\)
6. \(\begin{align}(\frac{-9}{6}+x)(b-\frac{-3}{6})& =(\color{red}{x}+\color{orange}{\frac{-3}{2}}).(\color{green}{b}+\color{blue}{\frac{1}{2}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(b)} + \color{red}{(x)} . \color{blue}{(\frac{1}{2})} + \color{orange}{(\frac{-3}{2})} . \color{green}{(b)} + \color{orange}{(\frac{-3}{2})} . \color{blue}{(\frac{1}{2})}& \text{Boogjes en uitschrijven} \\ & = bx+\frac{1}{2}x-\frac{3}{2}b-\frac{3}{2}.\frac{1}{2}& \text{Uitwerken } \\ & =bx+\frac{1}{2}x-\frac{3}{2}b-\frac{3}{4} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
7. \(\begin{align}(p-\frac{18}{20})(\frac{-5}{19}+q)& =(\color{red}{p}+\color{orange}{\frac{-9}{10}}).(\color{green}{q}+\color{blue}{\frac{-5}{19}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(q)} + \color{red}{(p)} . \color{blue}{(\frac{-5}{19})} + \color{orange}{(\frac{-9}{10})} . \color{green}{(q)} + \color{orange}{(\frac{-9}{10})} . \color{blue}{(\frac{-5}{19})}& \text{Boogjes en uitschrijven} \\ & = pq-\frac{5}{19}p-\frac{9}{10}q+\frac{9}{10}.\frac{5}{19}& \text{Uitwerken } \\ & =pq-\frac{5}{19}p-\frac{9}{10}q+\frac{45}{190} & \text{Uitrekenen (Geen Smoothie!)} \\ & = pq-\frac{5}{19}p-\frac{9}{10}q+\frac{9}{38} & \text{Vereenvoudig} \\\end{align}\)
8. \(\begin{align}(\frac{20}{13}+x)(\frac{-11}{10}+x)& =(\color{red}{x}+\color{orange}{\frac{20}{13}}).(\color{green}{x}+\color{blue}{\frac{-11}{10}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(x)} + \color{red}{(x)} . \color{blue}{(\frac{-11}{10})} + \color{orange}{(\frac{20}{13})} . \color{green}{(x)} + \color{orange}{(\frac{20}{13})} . \color{blue}{(\frac{-11}{10})}& \text{Boogjes en uitschrijven} \\ & = xx-\frac{11}{10}x+\frac{20}{13}x-\frac{20}{13}.\frac{11}{10}& \text{Uitwerken } \\ & =x^2+((\frac{-11}{10})+\frac{20}{13})x-(\frac{-220}{130}) & \text{Neem samen} \\ &=x^2+\frac{57}{130}x-(\frac{-220}{130}) & \text{Maak de som} \\ &=x^2+\frac{57}{130}x+\frac{22}{13} & \text{Vereenvoudig indien mogelijk 1} \\\end{align}\)
9. \(\begin{align}(p-\frac{-4}{15})(\frac{19}{3}+x)& =(\color{red}{p}+\color{orange}{\frac{4}{15}}).(\color{green}{x}+\color{blue}{\frac{19}{3}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(x)} + \color{red}{(p)} . \color{blue}{(\frac{19}{3})} + \color{orange}{(\frac{4}{15})} . \color{green}{(x)} + \color{orange}{(\frac{4}{15})} . \color{blue}{(\frac{19}{3})}& \text{Boogjes en uitschrijven} \\ & = px+\frac{19}{3}p+\frac{4}{15}x+\frac{4}{15}.\frac{19}{3}& \text{Uitwerken } \\ & =px+\frac{19}{3}p+\frac{4}{15}x+\frac{76}{45} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
10. \(\begin{align}(\frac{-6}{17}+a)(\frac{9}{12}+q)& =(\color{red}{a}+\color{orange}{\frac{-6}{17}}).(\color{green}{q}+\color{blue}{\frac{3}{4}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(q)} + \color{red}{(a)} . \color{blue}{(\frac{3}{4})} + \color{orange}{(\frac{-6}{17})} . \color{green}{(q)} + \color{orange}{(\frac{-6}{17})} . \color{blue}{(\frac{3}{4})}& \text{Boogjes en uitschrijven} \\ & = aq+\frac{3}{4}a-\frac{6}{17}q-\frac{6}{17}.\frac{3}{4}& \text{Uitwerken } \\ & =aq+\frac{3}{4}a-\frac{6}{17}q-\frac{18}{68} & \text{Uitrekenen (Geen Smoothie!)} \\ & = aq+\frac{3}{4}a-\frac{6}{17}q-\frac{9}{34} & \text{Vereenvoudig} \\\end{align}\)
11. \(\begin{align}((-6)+x)(q+\frac{-6}{8})& =(\color{red}{x}+\color{orange}{(-6)}).(\color{green}{q}+\color{blue}{\frac{-3}{4}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(q)} + \color{red}{(x)} . \color{blue}{(\frac{-3}{4})} + \color{orange}{(-6)} . \color{green}{(q)} + \color{orange}{(-6)} . \color{blue}{(\frac{-3}{4})}& \text{Boogjes en uitschrijven} \\ & = qx-\frac{3}{4}x-6q+6.\frac{3}{4}& \text{Uitwerken } \\ & =qx-\frac{3}{4}x-6q+\frac{18}{4} & \text{Uitrekenen (Geen Smoothie!)} \\ & = qx-\frac{3}{4}x-6q+\frac{9}{2} & \text{Vereenvoudig} \\\end{align}\)
12. \(\begin{align}(a+\frac{-1}{15})(10+q)& =(\color{red}{a}+\color{orange}{\frac{-1}{15}}).(\color{green}{q}+\color{blue}{10})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(q)} + \color{red}{(a)} . \color{blue}{(10)} + \color{orange}{(\frac{-1}{15})} . \color{green}{(q)} + \color{orange}{(\frac{-1}{15})} . \color{blue}{(10)}& \text{Boogjes en uitschrijven} \\ & = aq+10a-\frac{1}{15}q-\frac{1}{15}.10& \text{Uitwerken } \\ & =aq+10a-\frac{1}{15}q-\frac{10}{15} & \text{Uitrekenen (Geen Smoothie!)} \\ & = aq+10a-\frac{1}{15}q-\frac{2}{3} & \text{Vereenvoudig} \\\end{align}\)