Equations Inequalities and VIC's Question Bank Problem

[rkim81]

I don't know if I'm allowed to post these, but I disagree with question 14.
f(x)=ax^4-4x^2+ax-3
Therefore,
f(b)=ab^4-4b^2+ab-3
f(-b)=-ab^4+4b^2-ab-3
Therefore
f(b)-f(-b)=
(ab^4-4b^2+ab-3) -(-ab^4+4b^2-ab-3)
=ab^4-4b^2+ab-3 + ab^4-4b^2+ab+3
=2ab^4-8b^2+2ab Divide by 2
=ab^4-4b^2+ab

Please comment

[eddykim]

I don't have access to the original problem, but f(-b)=a(-b)^4+4(-b)^2-ab-3. The negatives don't matter because they're raised to an even power. So when f(-b) is subtracted, everything cancels out except the "ab" part. Is the answer listed 2ab?

[esledge]

I don't know if I'm allowed to post these, but I disagree with question 14.

You are allowed to post these. Here it is, for reference:

If f(x) = ax4 - 4x2 + ax - 3, then f(b) - f(-b) will equal:
(A) 0
(B) 2ab
(C) 2ab^4 - 8b^2 - 6
(D) -2ab^4 + 8b^2 + 6
(E) 2ab^4 - 8b^2 + 2ab - 6

f(x)=ax^4-4x^2+ax-3
Therefore,
f(b)=ab^4-4b^2+ab-3
f(-b)=-ab^4+4b^2-ab-3 [moderator note: should be +ab^4 - 4b^2]
Therefore
f(b)-f(-b)=
(ab^4-4b^2+ab-3) -(-ab^4+4b^2-ab-3)
=ab^4-4b^2+ab-3 + ab^4-4b^2+ab+3
=2ab^4-8b^2+2ab Divide by 2 [moderator note: Don't divide by 2, factor it to the side!]

Please comment

You made a sign error in the f(-b) calculation.
For example, ax^4 is a(-b)^4 = ab^4 because (-1)^4 = -1*-1*-1*-1 = 1.

And if this were an equation to solve, and every coefficient were a multiple of 2, then yes, divide by 2 to simplify. For example:
2x + 4y = 16 becomes x + 2y = 8.

But here, you don't have an equation. You don't know the value of the other side of the equal sign, just that it's f(b)-f(-b). When you divided by 2 at the last step, you got a result that would be (1/2)[f(b)-f(-b)].

[esledge]

I don't have access to the original problem, but f(-b)=a(-b)^4+4(-b)^2-ab-3. The negatives don't matter because they're raised to an even power. So when f(-b) is subtracted, everything cancels out except the "ab" part. Is the answer listed 2ab?

Good observation. The even exponents are the recognition skill tested by this question.

Yes, the answer is (B) 2ab.