Question:

Knowing that −1 is one of the roots of the polynomial P(x) = x³ − x² + x + 3, determine the sum of the modulus of the other roots.

Answer:

Roots: a, b, c (c = −1)

(x − a) (x − b) (x − c) = x³ − x² + x + 3

x³ − (a+b+c)x² + (ab+ac+bc)x − abc = x³ − x² + x + 3

Sum of roots:

a + b + c = 1

a + b − c = 1

a + b = 2

b = 2 − a

Product of roots:

abc = −3

ab(−1) = −3

ab = 3

a(2−a) = 3

2a − a² = 3

a² − 2a = −3

a² − 2a + 1 = −3 + 1

(a − 1)² = −2

a − 1 = ± i√2

a = 1 ± i√2

b = 1 ∓ i√2

Sum of the modulus of the other roots

= |1 + i√2| + |1 − i√2|

= √3 + √3

= 2√3

OMG same man!