If z = (x + y)(^4) – (x – y)(^4), which of the following is equal to one half of one quarter of z?

A. x(^3)y(^3)
B. 8x(^3)y(^3)
C. x(^3)y + xy(^3)
D. 8x(^3)y + 8xy(^3)
E. x(^3)y – xy(^3)


Does anyone have any suggestions on how to quickly solve this problem? The answer explanation instructs one to multiply everything out following the known quadratic equation pattern (x(^2) - y(^2)) = (x+y)(x-y). I decided to plug in values x=2, y=3 and came up with the correct answer C. Is this an unsafe method or is it only okay to use small prime numbers for VIC's? For instance if one chose x=0, y=1 then the answer would incorrectly be choice E.

x=2, y =3

(5)(^4) - (-1)(^4) = 625 - 1 = 624. (1/8)624= 78 Target answer

C. 2^3 (3) + (2) 3^3 = (24) + 2(27) = 78 Target answer

I guess for this problem, plugging in numbers will take longer than solving the expression, as you to plug in numbers for Expression as well the ans choices..

z = (x + y)(^4) – (x – y)(^4),

They are asking for 1/2 . 1/4 . Z = 1/8 Z

= 1/8 [(x + y)(^4) – (x – y)(^4) ]

as you said we can apply the a2 - b2 formula

= 1/8 [ (x+y)2 +(x-y)2] [(x+y)2 -(x-y)2]

= 1/8 . [ x2 +y2 +2xy +x2 +y2 - 2xy] [x2 +y2 +2xy -x2 -y2 + 2xy]

= 1/8. [2x2+2y2] 2x.2y

= xy(x2+y2)

= x3y + xy3

= C

if you take a quick glance at the answer choices, you'll notice that they're all different if you substitute 1 for both x and y. ordinarily, that's not an advisable substitution - precisely because it tends to create coincidences between answer choices - but it works out fine in this problem:
(a) 1
(b) 8
(c) 2
(d) 16
(e) 0
these values are easy to figure out by inspection, because you can ignore all the variables that are equal to 1 (i.e., multiplication by 1 can be ignored).

so, just plug 1 and 1 into the question prompt: z = 2^4 - 0^4, or 16. therefore, 1/2 of 1/4 of z is 2.
answer = c.

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notice the backwards order here; normally we solve the problem first and then investigate the answer choices, but, here, we've investigated the answer choices first. the reason is because the choices propounded (1 and 1) pose a very high risk of creating duplicate answers, so we should check out what happens to the answer choices before looking at the problem.