#[Distributive Law for Real Numbers][Ley distributiva para los números reales]#
#[If $a,$ $b,$ and $c$ are any real numbers, then:][Si $a, b,$ y $c$ son cualquieras números reales, entonces:]#
| #[Law][Ley]# |
Ejemplos |
$a(b \pm c) = ab \pm ac$
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$x(x+1)$\t = $x(x) + x(1) $ \\ \t ${}=x^2 + x$
$x^3(y-x)$\t ${}= x^3(y) - x^3(x) $ \\ \t ${}=x^3y - x^4$
$-7(x+y+z)$
${}= (-7)x + (-7)y + (-7)z$
${}=-7x - 7y - 7z$
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$(a \pm b)c = ac \pm bc$
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$(x+1)y$\t ${}=x(y) + 1(y) $ \\ \t ${}=xy + y$
$(1-3x+x^2)x$
${}=1(x) - 3x(x) + x^2(x)$
${}=x - 3x^2 + x^3$
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If what we are distributing is itself a sum or difference, then it is necessary to apply the distributive law more than once:
Examples
$\color{crimson}{(2x+1)}(3x-2)$ \t ${}=\color{crimson}{(2x+1)}(3x)\ \ + \ \ \color{crimson}{(2x+1)}(-2)$ \gap[40] \t
\\ \t ${}=6x^2 + 3x \ \ - \ 4x - 2$ \t
\\ \t ${}=6x^2 - x - 2$ \t
\\
\\ $\color{crimson}{(1 - y)}(1 + y - y^2)$ \t ${}=\color{crimson}{(1 - y)}(1)\ \ + \ \ \color{crimson}{(1 - y)}(y)\ \ - \ \ \color{crimson}{(1 - y)}(y^2)$ \t
\\ \t ${}=1 - y\ \ + \ \ y - y^2\ \ - \ \ (y^2 - y)$ \t
\\ \t ${}=1 - y \ \ + \ \ y - y^2\ \ - \ \ y^2 + y$ \t
\\ \t ${}=1 + y - 2y^2$ \t