Answer to Question #114746 in Algebra for ANJU JAYACHANDRAN

Question #114746
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.
1
Expert's answer
2020-05-28T17:35:45-0400


x5 - ax2 - ax + 1 =

x+ 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 polynomial has -1 as a root

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

Answer: a = -5.


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