Answer to Question #103605 in Algebra for ea

Question

Find the value of a for which the polynomial x^5 - ax^2 - ax +1 has −1 as a root with
multiplicity at least 2.

Answer 1

x⁵ - ax² - ax + 1 =

x⁵ + x⁴ - x⁴ - x³ - ax² - ax + x³ + x² - x² - x + x + 1 =

x⁴(x + 1) - x³(x + 1) - ax(x + 1) + x²(x + 1) - x(x + 1) + x + 1 =

(x⁴ - x³ + x² - (a + 1)x + 1)(x + 1) =

(x⁴ + x³ - 2x³ - 2x² + 3x² + 3x - (a + 4)x - (a + 4) + a + 4 + 1)(x + 1) =

(x³(x + 1) - 2x²(x + 1) + 3x(x + 1) - (a + 4)(x + 1) + (a + 5))(x + 1) =

(x³ - 2x² + 3x - (a + 4) + (a + 5)/(x + 1))(x + 1)²,

so we can see that the polynom has -1 as a root

with multiplicity at least 2 only if a + 5 = 0 or a = -5.

Answer: a = -5.