Polynomial Divison and Remainder Theorem Problem

[KarlyD]

12. Verify that the remainder theorem holds for the polynomial given below; that is, compute the remainder first using long division, and then using the remainder theorem.

f(x) = x^3 - 4x^2 + 5x - 3 : f(5)

My work:

Q(x)=quotient of the divison
P(x)=dividend
D(x)=divisor
R(x)=remainder

P(x)=Q(x)(x-a)+R(x)

P(5) = Q(5)^3 - 4(5)^2 + 5(5)-3

5(5)-3 is the remainder polynomial (with a value of 22)

**I think I did the remainder theorem aspect properly, but I can't wrap my head around how to go about dividing this by long divison. In my textbook, it says D(x) is the divisor, but I don't see how that relates to my problem. I have already researched it via internet and my textbooks (as I always do, before coming here), and can't find anything to help me!

 

[skeeter]

(x³ - 4x² + 5x - 3)/(x - 5) = x² + x + 10 + 47/(x - 5)

the remainder is 47 ... f(5) = 47

 

[KarlyD]

Where did you get (x-5) from?

 

[skeeter]

Where did you get (x-5) from?

The remainder theorem states that if a polynomial f(x) is divided by the linear function (x - k), the remainder is the value of f(k).

 

[KarlyD]

Thanks!