I am trying to write this rational expression in its simplest form:
2-x+2x^2-x^3/x^2-4
However I have no idea how to factor the four numbers on the top. I"m used to factoring just three numbers. Can you show me how to do this? Thanks!!!
use grouping
2-x+2x^2-x^3
= (2-x) + x^2(2-x)
= (2-x)(1+x^2)
so
(2-x+2x^2-x^3)/(x^2-4)
= (2-x)(1+x^2)/[(x-2)(x+2)]
= -(1+x^2)/(x+2)
2x+4x-3x/2x-4=
6x-3x/2x-4= 5/10x
To factor the numerator of the rational expression 2 - x + 2x^2 - x^3, we can use the factoring by grouping method.
1. Start by rearranging the terms in descending order of their exponents:
-x^3 + 2x^2 - x + 2.
2. Group the terms in pairs:
(-x^3 + 2x^2) + (-x + 2).
3. Factor out the greatest common factor from each pair:
x^2(-x + 2) + (-1)(-x + 2).
4. Notice that we now have a common binomial factor, (-x + 2), in both terms.
5. Factor out the common binomial factor:
(x^2 - 1)(-x + 2).
Now, let's simplify the rational expression by factoring the denominator.
To factor the denominator, x^2 - 4, we can use the difference of squares formula.
1. Rewrite the denominator as x^2 - 2^2, where 2 is the square root of 4.
2. Apply the difference of squares formula:
x^2 - 2^2 = (x - 2)(x + 2).
Now that both the numerator and denominator are factored, we can write the rational expression in its simplest form:
(2 - x + 2x^2 - x^3)/(x^2 - 4) = (x^2 - 1)(-x + 2)/((x - 2)(x + 2)).
Remember to always check for common factors and cancel out any common factors if possible. In this case, there are no common factors to cancel out, so the expression cannot be simplified further.