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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

 Feb 6, 2020
 #1
avatar+109345 
+1

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)

 

 

cool cool cool

 Feb 6, 2020
 #2
avatar+287 
+1

Thank you, CPhill!

 Feb 6, 2020

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