Question

If (x+k) is a factor of the polynomial x²-2x-15 and x³+a.find k and a

plz answer fast I will give 100 pts

Answer 1

$\huge\mathfrak\green{Heyaa!!}$

$\huge\mathfrak\red{Answer:-}$

Given polynomial: x²-2x-15

Therefore,

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.

Other given polynomial: x³+a

Therefore,

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

Answer 2

$\huge\rm\underline\purple{Solution:-}$

This mean that when (x+k) is divided by x² -2x -15 remainder is zero

By factor theorem ,

f(-k) = (-k)² -2(-k) - 15 = 0

=> f(-k) = k² + 2k -15 = 0

=> k² + 5k -3k -15 = 0

=> k(k+5) -3(k+5) = 0

=> (k+5)(k-3) = 0

=> k = -5 (or) k = 3

and (x+k) is also a factor of x³ + a

If k = 3 ,

(x + 3 ) is a factor of x³ + a

f(-3) = 0

=> (-3)³ + a = 0

=> a - 27 = 0

=> a = 27

If k = -5 , Then ( x -5 ) is factor of x³ + a

f(5) = 0

=> (5³) + a = 0

=> 125 + a = 0

=> a = -125

∴ k = -5 (or) k = 3

a = 27 (or) a = -125