Question 1014687: Put the expression ((1+1/x)/(2-5/x)) into the form ((a(x))/(b(x))for polynomials a(x) and b(x)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! a(x) = (1 + 1/x)
b(x) = (2 - 5/x)
c(x) = a(x) / b(x)
that would do it if they were not polynomials, but just functions.
however, .....
a polynomial has no variable in the denominator.
this is because a polynomial function must have non-negative integer exponents.
x^2.5 is not allowed because 2.5 is not an integer.
1/x is not allowed because 1/x is the same as x^-1 and -1 is not non-negative.
so, if i understand the problem correctly, we have to convert this expression into one where the variable x is not in the denominator.
stsrt with (1 + 1/x) / (2 - 5/x)
multiply numerator and denominator of this expression by x/x.
(1 + 1/x) * x = x + 1
(2 - 5/x) * x = 2x - 5
your new expression is (x + 1) / (2x - 5)
(x + 1) is a polynomial.
(2x - 5) is a polynomial.
we'll set a(x) equal to (x + 1)
we'll set b(x) equal to (2x - 5)
we'll set c(x) equal to a(x) / b(x).
that's your solution.
c(x) = a(x) / b(x)
a(x) = (x + 1)
b(x) = (2x - 5)
you can confirm that the new expression of c(x) will give you the same answer as your original expression by simply assigning a value to x randomly and then evaluating both the original expression and the final expression to see if they provide you with the same answer.
i did for x = 5 and for x = 36 and the answer was the same both times, so i'm reasonably sure that this is correct.
when you multiply a fraction by x/x, you are really multiplying the fraction by 1 because x/x is equivalent to 1, so the value of the fraction is not changed, even though it looks different.
1 + 1/x becomes x + 1
2 - 5/x becomes 2x - 5
each individual expression in the numerator and denominator don't give you the same answer, but the fraction does.
if x = 2, then 1 + 1/x equals 3/2 in the numerator of the original expression and x + 1 = 3 in the numerator of the revised expression.
if x = 2, then 2 - 5/x equals -1/2 in the denominator of the original expression and 2x - 5 = -1 in the denominator of the revised expression.
the fractions becomes (3/2) / (-1/2) in the original expression, and 3 / (-1) in the revised expression.
the numerators om the original expression and the revised expression are not the same, and the denominators in the original expression and the revised expression are not the same, but the original expression is equal to the revised expression, because .....
the original expression is equal to (3/2) / (-1/2) which is equal to (3/2) * (-2) which is equal to 3, and .....
the revised expression is equal to 3 / (-1) which is also equal to 3.
this says that the original expression is equivalent to the revised expression.
so, the solution to your problem is:
a(x) / b(x) = (x + 1) / (2x - 5)
both the numerator and the denominator are polynomials.
note that, while a(x) and b(x) meet the requirement of a polynomial, c(x) does not, because c(x) has a variable in the denominator.
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