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

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Werk uit en vereenvoudig

1. \((13+a)(q+\frac{12}{14})\)
2. \((\frac{-12}{3}+q)(\frac{-6}{12}-p)\)
3. \((\frac{14}{14}+x)(x+\frac{-19}{6})\)
4. \((p-\frac{-15}{19})(\frac{13}{20}+p)\)
5. \((b+\frac{13}{12})(p+\frac{-12}{12})\)
6. \((a-\frac{6}{4})(\frac{18}{13}+b)\)
7. \((b-6)((-6)-p)\)
8. \((\frac{-11}{3}+b)(\frac{1}{3}-b)\)
9. \((\frac{-8}{20}+x)(x-\frac{-14}{16})\)
10. \((q-\frac{7}{6})(a+\frac{18}{4})\)
11. \((\frac{-7}{6}+p)(\frac{12}{14}-p)\)
12. \((b+8)(a+\frac{-12}{13})\)

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Werk uit en vereenvoudig

Verbetersleutel

1. \(\begin{align}(13+a)(q+\frac{12}{14})& =(\color{red}{a}+\color{orange}{13}).(\color{green}{q}+\color{blue}{\frac{6}{7}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(q)} + \color{red}{(a)} . \color{blue}{(\frac{6}{7})} + \color{orange}{(13)} . \color{green}{(q)} + \color{orange}{(13)} . \color{blue}{(\frac{6}{7})}& \text{Boogjes en uitschrijven} \\ & = aq+\frac{6}{7}a+13q+13.\frac{6}{7}& \text{Uitwerken } \\ & =aq+\frac{6}{7}a+13q+\frac{78}{7} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
2. \(\begin{align}(\frac{-12}{3}+q)(\frac{-6}{12}-p)& =(\color{red}{q}+\color{orange}{(-4)}).(\color{green}{-p}+\color{blue}{\frac{-1}{2}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(-p)} + \color{red}{(q)} . \color{blue}{(\frac{-1}{2})} + \color{orange}{(-4)} . \color{green}{(-p)} + \color{orange}{(-4)} . \color{blue}{(\frac{-1}{2})}& \text{Boogjes en uitschrijven} \\ & = -pq-\frac{1}{2}q+4p+4.\frac{1}{2}& \text{Uitwerken } \\ & =-pq-\frac{1}{2}q+4p+\frac{4}{2} & \text{Uitrekenen (Geen Smoothie!)} \\ & = -pq-\frac{1}{2}q+4p+2 & \text{Vereenvoudig} \\\end{align}\)
3. \(\begin{align}(\frac{14}{14}+x)(x+\frac{-19}{6})& =(\color{red}{x}+\color{orange}{1}).(\color{green}{x}+\color{blue}{\frac{-19}{6}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(x)} + \color{red}{(x)} . \color{blue}{(\frac{-19}{6})} + \color{orange}{(1)} . \color{green}{(x)} + \color{orange}{(1)} . \color{blue}{(\frac{-19}{6})}& \text{Boogjes en uitschrijven} \\ & = xx-\frac{19}{6}x+1x-1.\frac{19}{6}& \text{Uitwerken } \\ & =x^2+((\frac{-19}{6})+1)x-(\frac{-19}{6}) & \text{Neem samen} \\ &=x^2-\frac{13}{6}x-(\frac{-19}{6}) & \text{Maak de som} \\\end{align}\)
4. \(\begin{align}(p-\frac{-15}{19})(\frac{13}{20}+p)& =(\color{red}{p}+\color{orange}{\frac{15}{19}}).(\color{green}{p}+\color{blue}{\frac{13}{20}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(p)} + \color{red}{(p)} . \color{blue}{(\frac{13}{20})} + \color{orange}{(\frac{15}{19})} . \color{green}{(p)} + \color{orange}{(\frac{15}{19})} . \color{blue}{(\frac{13}{20})}& \text{Boogjes en uitschrijven} \\ & = pp+\frac{13}{20}p+\frac{15}{19}p+\frac{15}{19}.\frac{13}{20}& \text{Uitwerken } \\ & =p^2+(\frac{13}{20}+\frac{15}{19})p+\frac{195}{380} & \text{Neem samen} \\ &=p^2+\frac{547}{380}p+\frac{195}{380} & \text{Maak de som} \\ &=p^2+\frac{547}{380}p+\frac{39}{76} & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)
5. \(\begin{align}(b+\frac{13}{12})(p+\frac{-12}{12})& =(\color{red}{b}+\color{orange}{\frac{13}{12}}).(\color{green}{p}+\color{blue}{(-1)})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(p)} + \color{red}{(b)} . \color{blue}{(-1)} + \color{orange}{(\frac{13}{12})} . \color{green}{(p)} + \color{orange}{(\frac{13}{12})} . \color{blue}{(-1)}& \text{Boogjes en uitschrijven} \\ & = bp-1b+\frac{13}{12}p-\frac{13}{12}.1& \text{Uitwerken } \\ & =bp-1b+\frac{13}{12}p-\frac{13}{12} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
6. \(\begin{align}(a-\frac{6}{4})(\frac{18}{13}+b)& =(\color{red}{a}+\color{orange}{\frac{-3}{2}}).(\color{green}{b}+\color{blue}{\frac{18}{13}})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(a)} . \color{green}{(b)} + \color{red}{(a)} . \color{blue}{(\frac{18}{13})} + \color{orange}{(\frac{-3}{2})} . \color{green}{(b)} + \color{orange}{(\frac{-3}{2})} . \color{blue}{(\frac{18}{13})}& \text{Boogjes en uitschrijven} \\ & = ab+\frac{18}{13}a-\frac{3}{2}b-\frac{3}{2}.\frac{18}{13}& \text{Uitwerken } \\ & =ab+\frac{18}{13}a-\frac{3}{2}b-\frac{54}{26} & \text{Uitrekenen (Geen Smoothie!)} \\ & = ab+\frac{18}{13}a-\frac{3}{2}b-\frac{27}{13} & \text{Vereenvoudig} \\\end{align}\)
7. \(\begin{align}(b-6)((-6)-p)& =(\color{red}{b}+\color{orange}{(-6)}).(\color{green}{-p}+\color{blue}{(-6)})& \text{Vereenvoud2g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(-p)} + \color{red}{(b)} . \color{blue}{(-6)} + \color{orange}{(-6)} . \color{green}{(-p)} + \color{orange}{(-6)} . \color{blue}{(-6)}& \text{Boogjes en uitschrijven} \\ & = -bp-6b+6p+6.6& \text{Uitwerken } \\ & =-bp-6b+6p+36 & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)
8. \(\begin{align}(\frac{-11}{3}+b)(\frac{1}{3}-b)& =(\color{red}{b}+\color{orange}{\frac{-11}{3}}).(\color{green}{-b}+\color{blue}{\frac{1}{3}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(-b)} + \color{red}{(b)} . \color{blue}{(\frac{1}{3})} + \color{orange}{(\frac{-11}{3})} . \color{green}{(-b)} + \color{orange}{(\frac{-11}{3})} . \color{blue}{(\frac{1}{3})}& \text{Boogjes en uitschrijven} \\ & = -bb+\frac{1}{3}b+\frac{11}{3}b-\frac{11}{3}.\frac{1}{3}& \text{Uitwerken } \\ & =-b^2+(\frac{1}{3}+\frac{11}{3})b-\frac{11}{9} & \text{Neem samen} \\ &=-b^2+\frac{12}{3}b-\frac{11}{9} & \text{Maak de som} \\ &=-b^2+4b-\frac{11}{9} & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)
9. \(\begin{align}(\frac{-8}{20}+x)(x-\frac{-14}{16})& =(\color{red}{x}+\color{orange}{\frac{-2}{5}}).(\color{green}{x}+\color{blue}{\frac{7}{8}})& \text{Vereenvoud3g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(x)} . \color{green}{(x)} + \color{red}{(x)} . \color{blue}{(\frac{7}{8})} + \color{orange}{(\frac{-2}{5})} . \color{green}{(x)} + \color{orange}{(\frac{-2}{5})} . \color{blue}{(\frac{7}{8})}& \text{Boogjes en uitschrijven} \\ & = xx+\frac{7}{8}x-\frac{2}{5}x-\frac{2}{5}.\frac{7}{8}& \text{Uitwerken } \\ & =x^2+(\frac{7}{8}+(\frac{-2}{5}))x-(\frac{-14}{40}) & \text{Neem samen} \\ &=x^2+\frac{19}{40}x-(\frac{-14}{40}) & \text{Maak de som} \\ &=x^2+\frac{19}{40}x+\frac{7}{20} & \text{Vereenvoudig indien mogelijk 1} \\\end{align}\)
10. \(\begin{align}(q-\frac{7}{6})(a+\frac{18}{4})& =(\color{red}{q}+\color{orange}{\frac{-7}{6}}).(\color{green}{a}+\color{blue}{\frac{9}{2}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(q)} . \color{green}{(a)} + \color{red}{(q)} . \color{blue}{(\frac{9}{2})} + \color{orange}{(\frac{-7}{6})} . \color{green}{(a)} + \color{orange}{(\frac{-7}{6})} . \color{blue}{(\frac{9}{2})}& \text{Boogjes en uitschrijven} \\ & = aq+\frac{9}{2}q-\frac{7}{6}a-\frac{7}{6}.\frac{9}{2}& \text{Uitwerken } \\ & =aq+\frac{9}{2}q-\frac{7}{6}a-\frac{63}{12} & \text{Uitrekenen (Geen Smoothie!)} \\ & = aq+\frac{9}{2}q-\frac{7}{6}a-\frac{21}{4} & \text{Vereenvoudig} \\\end{align}\)
11. \(\begin{align}(\frac{-7}{6}+p)(\frac{12}{14}-p)& =(\color{red}{p}+\color{orange}{\frac{-7}{6}}).(\color{green}{-p}+\color{blue}{\frac{6}{7}})& \text{Vereenvoud4g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(p)} . \color{green}{(-p)} + \color{red}{(p)} . \color{blue}{(\frac{6}{7})} + \color{orange}{(\frac{-7}{6})} . \color{green}{(-p)} + \color{orange}{(\frac{-7}{6})} . \color{blue}{(\frac{6}{7})}& \text{Boogjes en uitschrijven} \\ & = -pp+\frac{6}{7}p+\frac{7}{6}p-\frac{7}{6}.\frac{6}{7}& \text{Uitwerken } \\ & =-p^2+(\frac{6}{7}+\frac{7}{6})p-\frac{42}{42} & \text{Neem samen} \\ &=-p^2+\frac{85}{42}p-\frac{42}{42} & \text{Maak de som} \\ &=-p^2+\frac{85}{42}p-1 & \text{Vereenvoudig indien mogelijk 0} \\\end{align}\)
12. \(\begin{align}(b+8)(a+\frac{-12}{13})& =(\color{red}{b}+\color{orange}{8}).(\color{green}{a}+\color{blue}{\frac{-12}{13}})& \text{Vereenvoud1g (verplicht!) en herschik (niet verplicht)} \\& = \color{red}{(b)} . \color{green}{(a)} + \color{red}{(b)} . \color{blue}{(\frac{-12}{13})} + \color{orange}{(8)} . \color{green}{(a)} + \color{orange}{(8)} . \color{blue}{(\frac{-12}{13})}& \text{Boogjes en uitschrijven} \\ & = ab-\frac{12}{13}b+8a-8.\frac{12}{13}& \text{Uitwerken } \\ & =ab-\frac{12}{13}b+8a-\frac{96}{13} & \text{Uitrekenen (Geen Smoothie!)} \\\end{align}\)