Page:Elementary algebra (1896).djvu/143

violating cyclic order by the use of arrangements such as $b-c$, $a-c$, $a-b$, or $a-c$, $b-a$, $b-c$. It will always be found that the work is rendered shorter and easier by following cyclic order from the beginning, and adhering to it throughout the question.

EXAMPLES XIII. h.

Find the value of

1. ${\frac {a}{\left(a-b\right)\left(a-c\right)}}+{\frac {b}{\left(b-c\right)\left(c-a\right)}}+{\frac {c}{\left(c-a\right)\left(c-b\right)}}$
2. ${\frac {b}{\left(a-b\right)\left(a-c\right)}}+{\frac {c}{\left(b-c\right)\left(c-a\right)}}+{\frac {a}{\left(c-a\right)\left(c-b\right)}}$
3. ${\frac {z}{\left(x-y\right)\left(x-z\right)}}+{\frac {x}{\left(y-z\right)\left(z-y\right)}}+{\frac {y}{\left(z-x\right)\left(z-y\right)}}$
4. ${\frac {y+z}{\left(x-y\right)\left(x-z\right)}}+{\frac {z+x}{\left(y-z\right)\left(y-z\right)}}+{\frac {x+y}{\left(z-x\right)\left(z-y\right)}}$
5. ${\frac {b-c}{\left(a-b\right)\left(a-c\right)}}+{\frac {c-a}{\left(b-c\right)\left(b-a\right)}}+{\frac {a-b}{\left(c-a\right)\left(c-b\right)}}$
6. ${\frac {x^{2}yz}{\left(x-y\right)\left(x-z\right)}}+{\frac {y^{2}zx}{\left(y-z\right)\left(y-x\right)}}+{\frac {z^{2}xy}{\left(z-x\right)\left(z-y\right)}}$
7. ${\frac {x^{2}yz}{\left(x-y\right)\left(x-z\right)}}+{\frac {y^{2}zx}{\left(y-z\right)\left(y-x\right)}}+{\frac {z^{2}xy}{\left(z-x\right)\left(z-y\right)}}$
8. ${\frac {p-a}{\left(p-q\right)\left(p-r\right)}}+{\frac {q-a}{\left(q-r\right)\left(q-p\right)}}+{\frac {r-a}{\left(r-p\right)\left(r-q\right)}}$
9. ${\frac {p+q-r}{\left(p-q\right)\left(p-r\right)}}+{\frac {q+r-p}{\left(q-r\right)\left(q-p\right)}}+{\frac {r+p-q}{\left(r-p\right)\left(r-q\right)}}$
10. ${\frac {a^{2}}{\left(a^{2}-b^{2}\right)\left(a^{2}-c^{2}\right)}}+{\frac {b^{2}}{\left(b^{2}-c^{2}\right)\left(b^{2}-a^{2}\right)}}+{\frac {c2}{\left(c^{2}-a^{2}\right)\left(c^{2}-b^{2}\right)}}$
11. ${\frac {x+y}{\left(p-q\right)\left(p-r\right)}}+{\frac {x+y}{\left(q-r\right)\left(q-p\right)}}+{\frac {x+y}{\left(r-p\right)\left(r-q\right)}}$
12. ${\frac {q+r}{\left(x-y\right)\left(x-z\right)}}+{\frac {r+p}{\left(y-z\right)\left(y-x\right)}}+{\frac {p+q}{\left(z-x\right)\left(z-y\right)}}$