A few questions about this math problem: Find the difference quotient and simplify your answer:?

Question

f(x)=x^3 + 3x, f(x+h) - f(x)/h, h is not equal to zero

1.When you FOIL (x+h)^3 what do you get?

2. Secondly, the book is showing the answer as "3x^2 + 3xh +h^2 + 3" I can not achieve this answer.

Answer 1

1) you get x^3 + 3x^2 h + 3h^2 x + h^3
(x+h)(x+h)(x+h)
foil the first two
(x^2 + xh +xh +h^2)(x+h)
(x^2 + 2xh +h^2)(x+h)

Foil out whole thing
(x^3 + 2x^2 h +xh^2+ x^2h + 2xh^2 +h^3)
(x^3 + 3x^2 h +3xh^2 +h^3)

should probably remember (x+y)^3 = x^3 + 3x^2y + 3y^2 x + y^3 or at least the first couple rows of the pascal triangle.

2)
Apply difference quotition
[(x+h)^3 + 3(x+h) - [x^3 + 3x] ] / h

Distribute negative
[(x+h)^3 + 3(x+h) - x^3 - 3x] ] / h

Expand
[x^3 + 3x^2 h + 3h^2 x + h^3 +3x +3h - x^3 - 3x] / h

Simplify
[3x^2 h + 3h^2 x + h^3+3h] / h

factor h out
[3x^2 + 3hx + h^2+3]h / h

cancel h
[3x^2 + 3hx + h^2+3]

Answer 2

Hi,
1. FOIL is for multiplying two binomials. To cube a binomial, you can do one of these:
a) FOIL it and multiply that result by (x+h)
b) Use the bonomial expansion method.
c) Use your TI-89 calculator. :-)
I'll use method a) to keep from having to do a lot of explaining on the other methods.
(x+h)^3 =
(x+h)² (x+h)
(x² +2xh +h²)(x+h)
=x^3 +2x²h +2xh² +x²h +2xh² +h^3
= x^3 +3x²h +3xh² +h^3

Now, let's do the difference formula:
f(x+h) - f(x)
----------------
h
(x^3 +3x²h +3xh² +h^3) +3(x + h) - (x^3+3x)
-----------------------------------------------------
h

(x^3 +3x²h +3xh² +h^3) + (3x+3h) -x^3 -3x
----------------------------------------------------
h

3x²h +3xh² +h^3 +3h
---------------------------
h

3x²+3xh +h² +3 (Divide by h)

That should be it. Hope this helps some.
FE