How do you factor out 16x to the fourth power minus 81?

Answer 1

The answer below does find one root, and finding roots is one way to discover factors. But it does not find the complete factorization. x = -3/2 is also a root of the stated 'equation'. So two of the factors are (2x - 3) and (2x + 3). Then long division could be used to find remaining factor(s).

Recall that a2 - b2 = (a + b)(a - b). Let a2 = 16x4, and b2 = 81.

so, a = 4x2 and b = 9.

So: 16x4 - 81 = (4x2 + 9)(4x2 - 9), and (4x2 - 9) factors to (2x + 3)(2x - 3).

So the factorization is: (4x2 + 9)(2x + 3)(2x - 3).

With complex: (2x + 3i)(2x - 3i)(2x + 3)(2x - 3), where i = sqrt(-1)

So the other two roots are: 3i/2 and -3i/2

Finding a root:

16X^4 - 81 = 0

add 81 to each side : 16X^4 = 81

divide both sides by 16, leave as fraction

X^4 = 81/16

take 4th root both sides

X = 4root(81/16)

this can be expressed as.........

X = 4root(81)/4root(16)

X = 3/2 [indicates (2x - 3) is one of the factors]

check

16(3/2)^4 - 81 = 0

16 * 81/16 - 81 = 0

0 = 0

checks