newton's formula for the sums of powers of roots?

For example ,
let f(x) = a0 x⁵ + a1 x⁴+ a2 x³ + a3 x² + a4 x + a5 = a0 (x - α1) (x - α2) (x - α3) (x - α4) (x - α5) ,

Note that
- a1/a0 = α1 + ... + α5 ,
a2/a0 = α1 α2 + ... + α4 α5 ,
- a3/a0 = α1 α2 α3 + ... + α3 α4 α5 ,
a4/a0 = α1 α2 α3 α4 + ... + α2 α3 α4 α5 ,
- a5/a0 = α1 α2 α3 α4 α5 , then

f(x) / (x - α1)
= a0 (x - α2) (x - α3) (x - α4) (x - α5)

= a0 x⁴+ a0(- α2 - α3 - α4 - α5)x³ + a0(α2 α3 + ... + α4 α5)x² + a0(- α2 α3 α4 - ... - α3 α4 α5)x
+ a0(α2 α3 α4 α5)

= a0 x⁴+ a0(a1/a0 + α1)x³ + a0(a2/a0 - α1(α2 + α3 + α4 + α5))x² + a0(a3/a0 + α1(α2 α3 + ... + α4 α5))x
+ a0(a4/a0 - α1(α2 α3 α4 + ... + α3 α4 α5))

= a0 x⁴+ (a0α1 + a1)x³ + (a2 - α1 a0(α2 + α3 + α4 + α5))x² + (a3 + α1 a0(α2 α3 + ... + α4 α5))x
+ (a4 - α1 (-a3 - α1 a0(α2 α3 + ... + α4 α5))

= a0 x⁴+ (a0α1 + a1)x³ + (a2 - α1 (- a0α1 - a1) )x² + (a3 + α1 a0(α2 α3 + ... + α4 α5))x
+ (a4 - α1 a0(α2 α3 α4 + ... + α3 α4 α5))

= a0 x⁴+ (a0α1 + a1)x³ + (a0α1² + a1α1 + a2)x² + (a3 + α1 (a0α1² + a1α1 + a2))x
+ (a4 - α1 a0(α2 α3 α4 + ... + α3 α4 α5))

= a0 x⁴+ (a0α1 + a1)x³ + (a0α1² + a1α1 + a2)x² + (a0α1³ + a1α1² + a2α1 + a3)x
+ (a4 - α1 (- a0α1³ - a1α1² - a2α1 - a3)

= a0 x⁴+ (a0α1 + a1)x³ + (a0α1² + a1α1 + a2)x² + (a0α1³ + a1α1² + a2α1 + a3)x
+ (a0α1⁴+ a1α1³ + a2α1² + a3α1 + a4)