Exercises 72 and 73: A basic fact of algebra states that c is a root of a polynomial f if and only if ƒ(x) = (x − c)g(x) for some polynomial g. We say that c is a multiple root if ƒ(x) = (x − c)²h(x), where h is a polynomial.

Use Exercise 72 to determine whether c = −1 is a multiple root.
(a) x⁵ + 2x⁴ − 4x³ − 8x² − x + 2
(b) x⁴ + x³ − 5x² − 3x + 2

Data From Exercise 72

Exercises 72 and 73: A basic fact of algebra states that c is a root of a polynomial ƒ if and only if ƒ(x) = (x − c)g(x) for some polynomial g. We say that c is a multiple root if ƒ(x) = (x − c)²h(x), where h is a polynomial.

Show that c is a multiple root of ƒ if and only if c is a root of both ƒ and ƒ'.