Multiplicación. Polinomios por monomios.

El alumno notará que mediante este proceso hemos separado en realidad el dividendo en dos partes, $x^2+7x$ y $11x+77$, y dividido cada parte por $x+7$, y que el cociente completo es la suma de los cocientes parciales $x$ y $11$. Así,

$$\begin{gathered} x^2+18x+77\&=x^2+7x+11x+77=(x^2+7x)+(11x+77). \\ \therefore \frac{x^2+18x+77}{x+7}\&=\frac{x^2+7x}{x+7}+\frac{11x+77}{x+7}=x+11. \end{gathered}$$

2. Divide $a^2-2ab+b^2$ entre $a-b$. $$\begin{gathered} r*2cr|l{a}^2\&-\&2ab\&+\&b^2\&a-b \\ \mathrm{\backslash cline}6-6a^2\&-\&ab\&\&\&a-b \\ \mathrm{\backslash cline}1-5\&-\&ab\&+\&\mathrm{\backslash multicolumn}1@r{b}^2 \\ \&-\&ab\&+\&\mathrm{\backslash multicolumn}1@r{b}^2 \\ \mathrm{\backslash cline}2-5 \end{gathered}$$

3. Divide $a^4-a{b}^3+b^4+2a^2{b}^2-a^{3}b$ by $a^2+b^2$.

Ordena según las potencias decrecientes de $a$.

$$\begin{gathered} r*4cr|l{a}^4\&-\&a^{3}b\&+\&2a^2{b}^2\&-\&a{b}^3\&+\&b^4\&a^2+b^2 \\ \mathrm{\backslash cline}10-10a^4\&\&\&+\&a^2{b}^2\&\&\&\&\&a^2-ab+b^2 \\ \mathrm{\backslash cline}1-9\&-\&a^{3}b\&+\&a^2{b}^2\&-\&a{b}^3\&+\&\mathrm{\backslash multicolumn}1@r{b}^4 \\ \&-\&a^{3}b\&\&\&-\&a{b}^3 \\ \mathrm{\backslash cline}2-9\&\&\&+\&a^2{b}^2\&\&\&+\&\mathrm{\backslash multicolumn}1@r{b}^4 \\ \&\&\&+\&a^2{b}^2\&\&\&+\&\mathrm{\backslash multicolumn}1@r{b}^4 \\ \mathrm{\backslash cline}4-9 \end{gathered}$$

4. Divide $10a^2{b}^2-20b^4-17a^{3}b+6a^4+a{b}^3$ by $2a^2-4b^2-3ab$.

Ordenar según las potencias descendentes de $a$. $$\begin{gathered} r*4cr|l6a^4\&-\&17a^{3}b\&+\&10a^2{b}^2\&+\&a{b}^3\&-\&20b^4\&2a^2-3ab-4b^2 \\ \mathrm{\backslash cline}10-106a^4\&-\&9a^{3}b\&-\&12a^2{b}^2\&\&\&\&\&3a^2-4ab+5b^2 \\ \mathrm{\backslash cline}1-9\&-\&8a^{3}b\&+\&22a^2{b}^2\&+\&a{b}^3\&-\&\mathrm{\backslash multicolumn}1@r20b^4 \\ \&-\&8a^{3}b\&+\&12a^2{b}^2\&+\&16a{b}^3 \\ \mathrm{\backslash cline}2-9\&\&\&\&10a^2{b}^2\&-\&15a{b}^3\&-\&\mathrm{\backslash multicolumn}1@r20b^4 \\ \&\&\&\&10a^2{b}^2\&-\&15a{b}^3\&-\&\mathrm{\backslash multicolumn}1@r20b^4 \\ \mathrm{\backslash cline}5-9 \end{gathered}$$

5. Divide $5x^3-3x^4-4x^2+1+x$ by $1+2x-3x^2$.

Arrange according to ascending powers of $x$. $$\begin{gathered} r*4cr|l*2cr1\&+\&x\&-\&4x^2\&+\&5x^3\&-\&3x^4\&1\&+\&2x\&-\&3x^2 \\ \mathrm{\backslash cline}10-141\&+\&2x\&-\&3x^2\&\&\&\&\&1\&-\&x\&+\&x^2 \\ \mathrm{\backslash cline}1-9\&-\&x\&-\&x^2\&+\&5x^3\&-\&\mathrm{\backslash multicolumn}1@r3x^4 \\ \&-\&x\&-\&2x^2\&+\&3x^3 \\ \mathrm{\backslash cline}2-9\&\&\&\&x^2\&+\&2x^3\&-\&\mathrm{\backslash multicolumn}1@r3x^4 \\ \&\&\&\&x^2\&+\&2x^3\&-\&\mathrm{\backslash multicolumn}1@r3x^4 \end{gathered}$$

6. Divide $a^3+b^3+c^3-3abc$ by $a+b+c$.

Ordénese según las potencias decrecientes de $a$. $$\begin{gathered} r*4crclcr|l{a}^3\&\&\&\&\&-\&3abc\&\&\&+\&b^3\&+\&c^3\&a+b+c \\ \mathrm{\backslash cline}14-14a^3\&+\&a^{2}b\&+\&a^{2}c\&\&\&\&\&\&\&\&\&[t]a^2\&-ab-ac \\ \&+b^2-bc+c^2 \\ \mathrm{\backslash cline}1-13\&-\&a^{2}b\&-\&a^{2}c\&-\&3abc\&\&\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&-\&a^{2}b\&-\&a{b}^2\&-\&abc \\ \mathrm{\backslash cline}2-13\&-\&a^{2}c\&+\&a{b}^2\&-\&2abc\&\&\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&-\&a^{2}c\&\&\&-\&abc\&-\&a{c}^2 \\ \mathrm{\backslash cline}2-13\&\&\&\&a{b}^2\&-\&abc\&+\&a{c}^2\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&\&\&\&a{b}^2\&\&\&\&\&+\&b^3\&+\&\mathrm{\backslash multicolumn}1@r{b}^{2}c \\ \mathrm{\backslash cline}4-13\&\&\&\&\&-\&abc\&+\&a{c}^2\&-\&b^{2}c\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&\&\&\&\&-\&abc\&\&\&-\&b^{2}c\&-\&\mathrm{\backslash multicolumn}1@rb{c}^2 \\ \mathrm{\backslash cline}6-13\&\&\&\&\&\&\&\&a{c}^2\&+\&b{c}^2\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \&\&\&\&\&\&\&\&a{c}^2\&+\&b{c}^2\&+\&\mathrm{\backslash multicolumn}1@r{c}^3 \\ \mathrm{\backslash cline}9-13 \end{gathered}$$

Ejercicio 24.

Divide:

1. $x^2+15x+56$ entre $x+7$.

2. $x^2-15x+56$ entre $x-7$.

3. $x^2+x-56$ por $x-7$.

4. $x^2-x-56$ por $x+7$.

5. $2a^2+11a+5$ entre $2a+1$.

6. $6a^2-7a-3$ por $2a-3$.

7. $4a^2+23a+15$ entre $4a+3$.

8. $3a^2-4a-4$ por $2-a$.

9. $x^4+x^2+1$ entre $x^2+x+1$.