What is the completely factored form of x⁴ + 8x² - 9?

Solution:

Given x⁴ + 8x² - 9

Let us use the u-substitution to factor this polynomial.

Let x² = u

It becomes u² + 8u - 9

We use quadratic formula to find the roots

m= -b ± √(b² - 4ac) / 2a

Here, a = 1, b = 8, c = -9

u = -8 ± √(8² - 4(-9)) / 2(1)

= -8 ± √(64 + 36) / 2

= -8 ± √(100) / 2

= -8 ± 10 / 2

= -18/2, 2/2

u = -9, 1

But u = x² = -9, 1

x = √-9, √1

x = ±3i, ±1

Factored form (x + 3i)(x - 3i)(x + 1)(x - 1)

What is the completely factored form of x⁴ + 8x² - 9?

Summary:

The completely factored form of x⁴ + 8x² - 9 is (x + 3i)(x - 3i)(x + 1)(x - 1).