simplify this expression...
x^4+125x/(x^3+5x^2-x-5)
I get x^2+5x+25/(-5)
but i cant get it to check...did i make an error somewehre?
To simplify the expression x^4 + 125x / (x^3 + 5x^2 - x - 5), follow these steps:
Step 1: Factor the denominator (x^3 + 5x^2 - x - 5). To do this, you can use synthetic division, polynomial long division, or through inspection. Let's use factoring by grouping:
First, group the terms:
(x^3 + 5x^2) + (-x - 5)
Next, factor out the common factors from each group:
x^2(x + 5) - 1(x + 5)
Now, you can factor out (x + 5) from both terms:
(x^2 - 1)(x + 5)
Step 2: Rewrite the expression using the factored form of the denominator:
x^4 + 125x / ((x^2 - 1)(x + 5))
Step 3: Factor the numerator. Notice that x^4 + 125x is a sum of cubes:
(x^3)^2 + (5x)^3
Using the sum of cubes formula (a^3 + b^3 = (a + b)(a^2 - ab + b^2)), you can rewrite the numerator as:
(x^3 + 5x)((x^3)^2 - (x^3)(5x) + (5x)^2)
Simplifying further:
(x^3 + 5x)(x^6 - 5x^4 + 25x^2)
Step 4: Cancel out common factors between the numerator and denominator. In this case, there is a common factor of (x^3 + 5x):
(x^3 + 5x)(x^6 - 5x^4 + 25x^2) / ((x^2 - 1)(x + 5))
Canceling the common factor of (x^3 + 5x):
(x^6 - 5x^4 + 25x^2) / (x^2 - 1)
Step 5: Simplify further. Notice that both the numerator and the denominator can be factored:
x^2(x^4 - 5x^2 + 25) / (x^2 - 1)
Further factorization is not possible for the numerator or the denominator without knowing the value of x. Therefore, the expression cannot be simplified any further.
As for your simplified expression of x^2 + 5x + 25 / (-5), it seems that you made an error. The correct simplified expression is x^2(x^4 - 5x^2 + 25) / (x^2 - 1).