1. ## find q(x)

(x) = x^5 - 7x^4 + 3x^3 - 2x^2 + 5x - 1. Then P(x) = (x - 2)Q(x) + R. If so, Q(x) =

These problems always give me a hard time, can anyone explain it to me? The answer is
x^4 - 5x^3 - 7x^2 - 16x - 27 but I have no idea how to get there.

2. Originally Posted by nandokommando
f(x) = x^5 - 7x^4 + 3x^3 - 2x^2 + 5x - 1. Then P(x) = (x - 2)Q(x) + R. If so, Q(x) =

These problems always give me a hard time, can anyone explain it to me? The answer is
x^4 - 5x^3 - 7x^2 - 16x - 27 but I have no idea how to get there.
The question is asking what multiplies by (x-2) to give f(x). It's the algebraic equivalent of saying what multiplies by 7 to give 28.

Use division: $\displaystyle Q(x) = \frac{f(x)}{(x-2)}$

I would use long division to solve but some prefer synthetic division (whatever that may be!)

You should be able to ignore the remainder R in this case

3. Originally Posted by nandokommando
(x) = x^5 - 7x^4 + 3x^3 - 2x^2 + 5x - 1. Then P(x) = (x - 2)Q(x) + R. If so, Q(x) =

These problems always give me a hard time, can anyone explain it to me? The answer is x^4 - 5x^3 - 7x^2 - 16x - 27 but I have no idea how to get there.
Hi nandokommando,

$\displaystyle P(x)=(x-2) \cdot Q(x) + R$ tells you that the dividend P(x) = the divisor (x-2) times the quotient Q(x) plus the remainder R. This is the Remainder Theorem.

Now, either use long division or synthetic division to determine the depressed polynomial Q(x) after dividing by (x - 2).

Synthetic Division:

Code:

2 ] 1  -7    3   -2    5    1
--      2  -10  -14  -32  -54
---------------------------
1  -5   -7  -16  -27  -53
The depressed polynomial is $\displaystyle x^4-5x^3-7x^2-16x-27+\frac{-53}{x-2}$