Use the Remainder Theorem to determine if (x-2) is a factor of . f(x)=x^4-2x^3+3x^2-10x+3

Question

Use the Remainder Theorem to determine if (x-2) is a factor of . f(x)=x^4-2x^3+3x^2-10x+3

Answer 1

To do this using the remainder theorem, and more specifically the factor theorem,  we can use synthetic division.

 

2 | 1  -2  3  -10  3   

          2  0    6  -8

     1   0  3   -4   -5      <---- since this last number, -5, is not 0, it means x - 2 is not a factor, as the theorem states that this number, the remainder, must be equal to 0.

 

If you can't follow what i did above, i simply brought down the first number, then multiplied it by our number 2, brought that number up, added -2 and 2 to get 0, multiplied it by 2, brought that 0 up, added it to 3,  and so on.

      

Answer 2

f(2) = (2)^4 - 2(2)^3 + 3(2)^2 - 10 (2) + 3

 

= 16 - 16 + 12 - 20 + 3

 

= -5

 

 

We can determine that (x-2) is not a factor of the given function because f(2) must be equal to zero if x-2 was a 

 

factor of f(x)