+296 1. When \(f(x) = ax^3 - 6x^2 + bx - 5\) is divided by \(x-1,\) the remainder is \(-5.\) When \(f(x)\) is divided by \(x+2,\) the remainder is \(-53.\) Find the ordered pair \((a, b).\)
2. When a polynomial is divided by \(-3x^5 + 10x - 11,\) what are the possible degrees of the remainder? Enter all the possible values, separated by commas.
Thank you guys :D
+128460 1.
Using the Remainder Theorem we have the following system
a(1)^3 - 6(1)^2 + b(1) - 5 = -5 ⇒ a - 6 + b = 0 ⇒ a + b = 6 (1)
And
a(-2)^3 - 6(-2)^2 + b(-2) - 5 = -53 ⇒ -8a - 24 - 2b - 5 = -53 ⇒ -8a - 2b = -24 ⇒
-4a - b = -12 (2)
Add (1) and (2) and we have that
-3a = -6
a = 2
And a + b = 6 so b = 4
So
(a, b) = ( 2 , 4)
