The trinomial factor of the sum of two perfect cubes is (16a^4 -12a^2 + 9). What is?

Question

The trinomial factor of the sum of two perfect cubes is (16a^4 -12a^2 + 9). What is
a)the binomial factor?
b)the original product

Answer 1

Let X³ and Y³ be both perfect cubes. You can say that

X³+ Y³ = (X+ Y)(X²- XY+ Y²)

Comparing the trinomial factor above with (16a^4- 12a^2+ 9), you can say that

X² = 16a^4----> X = 4a² or -4a²
XY = 12a²----> Y = 3 or -3
Y² = 9---> Y = 3 or -3

So if X equals 4a², Y must be 3. On the other hand, if X equals -4a², Y must be -3. So the perfect cubes you are looking for can be either (4a²)³ = 64a^6 and (3)³ = 27 OR (-4a²)³ = -64a^6 and -27.
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a)
The binomial factor is (X+ Y) which, according to what was said above, can be either (4a^2+ 3) or (-4a^2- 3).

b)
The original product therefore can be either
___ (4a^2+ 3)(16a^4- 12a^2+ 9)
or
___ (-4a^2- 3)(16a^4- 12a^2+ 9).

Answer 2

16a^4 - 12a^2 + 9 = (4a^2 - 6a + 3)(4a^2 + 6a + 3)