...EVANS, Madison University, Hamilton, NY x a by xc dy m Find x by quadratics. Since the sum of the fifth powers of two numbers is divisible by the sum of the numbers, both numerator and denominator of the first member of the given equation is divisible by 2a: + af-...

...(32 a5 - 243 65) H- (2 a - 3 6). CASE II. From these results it may be assumed that : 103. The sum of two equal odd powers of two numbers is divisible by the sum of the numbers. EXERCISE XXV. Write by inspection the results in the following examples : 1. (a;s + ys)-=-(a; + y)....

...— ax -f- a2, and \y = ж4 - ar'y + *y _ жу» + у4, x-тУ and so on, it follows that the sum of two equal odd powers of two numbers is divisible by the sum of the numbers. Ex. 4O. Resolve into factors : 1. a^ + y2. 6. 216a2 + 512c2. 2. arЧ-8. 7. 3. ^ + 216. 8. 4. ys + 64z2....

...r'y + 36 ^y2 - 24 .ту3 + 16y4. ' From these results it may be assumed that : 103. The sum of ¿wo equal odd powers of two numbers is divisible by the sum of the numbers. EXERCISE XXV. Write by inspection the results in the following examples : 1. (,f + f)-i.(x + y). 5....

...81a;4- 54^y + 36^y2- 24V+ 16y4. From these results it may be assumed that : 103. 7%e swra q/" <wo eywaZ odd powers of two numbers is divisible by the sum of the numbers. EXERCISE XXV. Write by inspection the results in the following examples ; 1. (a* + y3)-i-(x + y). 5....

...<¿±&. = a< _ aз¿ + a2¿2 — a¿8 + ¿4 a + о In general, it will be found that the sum of two like odd powers of two numbers is divisible by the sum of the numbers. Compare the quotients in (3) and (4) with those in (1) and (2). (5) -= = (6) ^~f = . > У IJ In general,...

...the sum of the numbers. x™ + y" is divisible by x + y if n be any odd whole number. (3) That is : the sum of any two equal odd powers of two numbers is always divisible by the sum of the numbers. x™ + »/" is never divisible by x + y or x — y, when...

...It may be shown by actual division that : ; and so on. - a+o a+b Hence, in general, it will be found that, The sum of any two equal odd powers of two numbers...34. Thus, in general, If a and b are coefficients, ayf+btf* divides а*з?г 4- bny"m (n being any odd and positive integer, also m and p are integral,...

...— y. x + y x + y уг, Eem., — , Eem., From the above we infer that the difference of the same powers of two numbers is divisible by the sum of the numbers only when tlje powers are even. 3. x — y x — y x — y = ж2 + xy + y2, Eem., 2 y3. = x3 + x'y...

...— з?у + yhf — xf + y4, Rem., — 2?A From the above we infer that the difference of the same powers of two numbers is divisible by the sum of the numbers only when the powers are even. 3. zr;r = a; + y, Rem., 2y*. у + ?/2, Rem., xy xt + yt xy ^,¿4 = *"...