How do you simplify # (3x^2 +4x -4) /( 3x^4 -2x^3)#?

Question

Genesis Allen · Accepted Answer

# (3x^2 +4x -4) /( 3x^4 -2x^3) = (x+2)/(x^3)#

After factoring the denominator and numerator, we cut whatever is present on each.

It's easy to find the denominator; just use the term with the highest commonality.

#3x^4 - 2x^3 = x^3(3x-2)#

The numerator is more complicated, let's assume that there are real numbers #a# and #b# such that #(3x+a)(x-b) = 3x^2 +4x -4#. (Or, we can put that 3 in evidence, and work with #(x+a)(x+b)# but then we'd have to work with rationals.)

By expanding that product we have that #(3x +a)(x+b) = 3x^2 +ax + 3bx + ab# #3x^2 +ax + 3bx + ab = 3x^2 + (a+3b)x + ab#

So we need to find two numbers that #a + 3b = 4# #ab = -4#

And that #f(-a/3)# and #f(-b)# are roots, i.e.: zero the function (you can check that by plugging either value on the parenthesis and seeing it will make a zero).

#a = -2# and #b = 2# are numbers that fit all three criterias. (It's a bit of a trial and error and that's why it's easier to do it this way only when you have nice numbers. If the going gets tough use the quadratic equation to find the roots, and test the positions.)

So know we have: # (3x^2 +4x -4) /( 3x^4 -2x^3) = ((3x-2)(x+2))/(x^3(3x-2))#

We cancel the common term # (3x^2 +4x -4) /( 3x^4 -2x^3) = (x+2)/(x^3)#

Eva Abramson · Answer

undefined To simplify the expression (3x^2 + 4x - 4) / (3x^4 - 2x^3), we can factor out the greatest common factor from both the numerator and the denominator. In this case, the greatest common factor is x^2. Factoring out x^2 from the numerator gives us x^2(3x + 4 - 4/x^2), and factoring out x^2 from the denominator gives us x^2(3x^2 - 2x).

Now, we can cancel out the x^2 terms in the numerator and denominator, leaving us with (3x + 4 - 4/x^2) / (3x^2 - 2x).

This is the simplified form of the expression (3x^2 + 4x - 4) / (3x^4 - 2x^3).