# Verify if 2x - 1/( x^2 + 1 ) = f(x) if f(x) = ( 2x^3 + 2x - 1 )/ ( x^2 + 1 )

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2x - 1/(x^2 + 1) = f(x)

f(x) = (2x^3+ 2x -1)/(x^2 + 1)

We will subsitute f(x) in the expression to verify :

==> 2x - 1/(x^2 + 1) = (2x^3+ 2x -1)/(x^2 + 1)

Now we will resrite the left side with a common denominator (x^2 + 1)

==> (2x(x^2+1) -1]/(x^2 +1) = (2x^3 + 2x -1)/(x^2 +1)

==> Now expand brackets:

==> (2x^3 + 2x -1)/(x^2 +1) = (2x^3 + 2x -1)/(x^2 + 1)

We notice that left side = right side

Then the expression is true.

==> 2x- 1/(x^2 + 1) = f(x)

To verify if 2x - 1/( x^2 + 1 ) = f(x) if f(x) = ( 2x^3 + 2x - 1 )/ ( x^2 + 1 ).

Given f(x) = (2x^3+2x-1)/(x^2+1)

We divide 2x^3+2x-1 by x^2+1:

x^2+1) 2x^3 +2x-1( 2x

2x^3 +2x

-----------------------------

-1.

_---------------------------

So if 2x^3 +2x-1 is actually divided by x^2+1, we get a quotient 2x and a remainder -1.

Therefore by division algorithm, we can write:

f(x) = (2x^3+2x-1)/(x^2+1) = 2x - 1/(x^2+1).

Thus we have verified the fact that if f(x) = 2x^2+2x-1, then

f(x) = 2x-1/(x^2+1).

To prove that 2x - 1/( x^2 + 1 ) = f(x), we'll substitute f(x) by the given expression ( 2x^3 + 2x - 1 )/ ( x^2 + 1 ).We could solve the problem in 2 ways: either we'll calculate the difference 2x - 1/( x^2 + 1 ), or we'll re-write the expression of f(x).

We'll re-write f(x):

( 2x^3 + 2x - 1 )/ (x^2 + 1) = (2x^3 + 2x)/(x^2 + 1) - 1/(x^2 + 1)

We'll factorize by 2x the first ratio:

(2x^3 + 2x)/(x^2 + 1) = 2x(x^2 + 1)/(x^2 + 1)

We'll simplify the first ratio:

2x(x^2 + 1)/(x^2 + 1) = 2x

**f(x) = 2x - 1/(x^2 + 1)**

**We can notice that the result or re-writing f(x) is similar to the original expression 2x - 1/(x^2 + 1).**