algebra 2 problem

Question

What is f in factored form for this f(x) = x^4 + 24x^2 +23

Answer 1

This quartic polynomial is a quadratic in x². Also, it is a monic polynomial (Leading coefficient 1). Therefore we find two numbers with sum 24 and product 23. These are 1 and 23.

 

x⁴ + 24x² + 23 = (x² + 1)(x² + 23)

Answer 2

x⁴ + 24x² + 23 = (x²)² + 24(x²) + 23

 

Let u = x² to obtain u² + 24u + 23 = (u + 23)(u + 1)

 

                                                    f(x) = (x² + 23)(x² + 1)