Question

how we can easily find the factors of x^4+2x^2+9

Answer 1

x⁴ + 2x² + 9 = 0 

x² = (−2 ± √(4-36) / 2 
x² = (−2 ± √32 i) / 2 
x² = (−2 ± 4√2 i) / 2 
x² = −1 ± 2√2 i 

x² = −1 + 2√2 i = (1 + √2 i)² -----> x = 1 + √2 i, x = −1 − √2 i 
x² = −1 − 2√2 i = (1 − √2 i)² -----> x = 1 − √2 i, x = −1 + √2 i 

Now we use roots, two at a time, to find quadratics that have these complex roots 

x = 1 ± √2 i 
x − 1 = ± √2 i 
(x − 1)² = 2i² 
x² − 2x + 1 = −2 
x² − 2x + 3 = 0 -----> x² − 2x + 3 is a factor of x⁴ + 2x² + 9 

x = −1 ± √2 i 
x + 1 = ± √2 i 
(x + 1)² = 2i² 
x² + 2x + 1 = −2 
x² + 2x + 3 = 0 -----> x² + 2x + 3 is a factor of x⁴ + 2x² + 9 


x⁴ + 2x² + 9 = (x² + 2x + 3) (x² − 2x + 3)