Simplify.
1. x^2 - 1/2x^2 - x - 1
2. x^2 - 16/x^2 - 6x + 8
3. 6x^2 + 4x/2x^2 + 4x
My answers:
1. ???
2. x+4/x-2
3. 3x?
For #1, i cannot figure out what 2x^2 - x - 1 factored is
I'm sure you can factor
x^2-x-2, no?
This is the same things
(2x+1)(x-1)
Not quite sure what it has to do with #1, though
#2 ok
#3. Since the top and bottom are of equal degree, there should be no unbalanced x's between the top and bottom
(6x^2+4x)/(2x^2+4x)
= 2x(3x+2) / 2x(x+2)
= (3x+2)/(x+2)
You can't just cancel the 2's. That's like saying that
8/7 = (6+2)/(5+2) = 6/5
thank you!
To simplify these expressions, we can combine like terms and use factoring or simplifying fractions where applicable. Here's how you could simplify each expression:
1. x^2 - 1/2x^2 - x - 1:
To combine like terms, we need to find a common denominator for the fraction term. In this case, the common denominator is 2. Multiplying the numerator and denominator of 1/2x^2 by 2 gives us 2/2x^2. Simplifying this gives us:
x^2 - 2/2x^2 - x - 1
Now, combining like terms:
x^2 - 2/2x^2 can be simplified to x^2 - x^2 = 0
So the expression simplifies to -x - 1.
2. x^2 - 16/x^2 - 6x + 8:
Since we have a fraction in the expression, we can simplify further by multiplying through by x^2 to eliminate the fraction:
x^2 * (x^2 - 16/x^2) - 6x * x^2 + 8*x^2
Expanding and simplifying each term:
x^4 - 16 - 6x^3 + 8x^2
No further simplification can be done, so the expression remains as it is: x^4 - 16 - 6x^3 + 8x^2.
3. 6x^2 + 4x/2x^2 + 4x:
Similar to the steps we took in expression 1, let's find a common denominator for the fraction term, which is 2x. Multiplying 4x by 2 gives us 8x, and multiplying 2x^2 by 2 gives us 4x^2:
(6x^2 + 4x) / (2x^2 + 4x)
Now, combining like terms:
6x^2 + 4x divided by 2x^2 + 4x
The common factor here is 2x, so we can simplify further:
(2x * (3x + 2)) / (2x * (x + 2))
Canceling out the common factor:
(3x + 2) / (x + 2)
So the expression simplifies to (3x + 2) / (x + 2).
In summary:
1. x^2 - 1/2x^2 - x - 1 simplifies to -x - 1.
2. x^2 - 16/x^2 - 6x + 8 remains as x^4 - 16 - 6x^3 + 8x^2.
3. 6x^2 + 4x/2x^2 + 4x simplifies to (3x + 2) / (x + 2).