Question

if (x+k) is a factor of the polynomial x^2-2x-15 and x^3+a . find k and a

Answer 1

Answer: k=-5 or 3

a=27 or -125

Sol:

Let f(x) = x2–2x–15,

(x+ k) is a factor of the polynomial x2–2x–15

⇒ f(-k) = 0

⇒ f(-k) = (-k)2–2(-k) –15 = 0

⇒ k2+ 2k –15 = 0

⇒ (k + 5)(k - 3) = 0

⇒ k = -5 or 3.

Let f(x) = x3+ a,

(x + k) is a factor of the polynomial x3+ a

⇒ (x + 3) is a factor of the polynomial x3+ a

⇒ f(-3) = 0

⇒ (-3)3+ a = 0

⇒ -27 + a = 0

⇒ a = 27

or

⇒ (x - 5) is a factor of the polynomial x3+ a

⇒ f(5) = 0

⇒ (5)3+ a = 0

⇒ 125 + a = 0

⇒ a = -125.

K= -5 or 3

a= 27 or -125

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