Who can accept the challenge of solving this question 4 me...?

Question

Given that f(a) = a^3 - ba^2 - 4b^2a + 4b^3. Show that a-2b is a factor of f(a). Find in terms of b the remainder when f(a) is divided by a+b.

Answer 1

Synthetic division is easiest.

2b | +01 -1b -4b² +4b³
----| +00 +2b +2b² -4b³
---------------------------
----| +01 +1b -2b² +00

So (a - 2b) divides evenly, leaving a² + ba - 2b²

Trying -b (a + b = 0, so a = -b):

-b | +01 -1b -4b² +4b³
----| +00 -1b +2b² +2b³
---------------------------
----| +01 -2b -2b² +6b³

So the remainder when divided by (a + b) would be 6b³.

Answer 2

"Given that f(a) = a^3 - ba^2 - 4b^2a + 4b^3.
Show that a-2b is a factor of f(a). "

f(2b) = (2b)^3 - b(2b)^2 - 4b^2(2b) + 4b^3 = 0
which means a-2b is a factor of f(a).



"Find in terms of b the remainder when f(a) is divided by a+b."

The remainder = f(-b) = (-b)^3 - b(-b)^2 - 4b^2(-b) + 4b^3 = 6b^3

Answer 3

f(a) = a^3 - ba^2 - 4b^2a + 4b^3.

In order for (a - 2b) to be a factor, f(2b) must be equal to 0. Let's see if that's the case.

f(2b) = [2b]^3 - b[2b]^2 - 4[b^2](2b) + 4b^3
f(2b) = 8b^3 - b[4b^2] - 8b^3 + 4b^3
f(2b) = 8b^3 - 4b^3 - 8b^3 + 4b^3

Which, as you can see, the terms cancel each other out, meaning
f(2b) = 0.

For that reason, (a - 2b) is a factor of f(a).

In order to find the remainder when f(a) is divided by (a + b), all we have to do is calculate f( [whatever a value makes a + b = 0]), or f(-b).

f(-b) = [-b]^3 - b[-b]^2 - 4b^2[-b] + 4b^3
f(-b) = -b^3 - b(b^2) + 4b^3 + 4b^3
f(-b) = -b^3 - b^3 + 4b^3 + 4b^3
f(-b) = -2b^3 + 8b^3
f(-b) = 6b^3

So the remainder is 6b^3

Answer 4

f(2b) = (2b)^3 - b(2b)^2 - 4b^2(2b) + 4b^3 = 0
which means a-2b is a factor of f(a).
f(-b) = (-b)^3 - b(-b)^2 - 4b^2(-b) + 4b^3 = 6b^3

Answer 5

I can't answer this question because im tired, maybe tomorrow.

Answer 6

Not me... I have a headache.