A few questions on simplifying rational expressions...
1). 15x^2/10x this simplifies to 3x^2/2 according to my book. But I would think it should be 3/2x since x^2-x=x. Why not?
2). Simplify 3xy/xy+x. How come the (xy)'s don't cancel so you are left with 3/x?
3). Can you break it down, step by step how (4y-8y^2)/(10y-5) becomes -4y/5?
Thank you!
1. You need to divide both numerator and denominator by the same term, in this case, 5x.
x^2-x = x(x-1) ≠ x
2. You need to divide both by the same term.
(xy+x)/xy = 1 + 1/y
3. Sorry, but I have to leave now.
1) To simplify the rational expression 15x^2 / 10x, you can start by factoring out the common term, which is 5x. This gives you: (5x * 3x) / (5 * 2x).
Then, you can cancel out the common factors in the numerator and denominator. The 5's in the numerator and denominator cancel out, as well as one of the x's. This leaves you with (3x) / (2), or 3x/2.
The reason it is written as 3x/2 and not 3/2x is because division is not commutative. In other words, the order matters when dividing. So, it is more conventional to write the constant first and the variable last.
2) In the expression 3xy / (xy+x), you can factor out the common term, which is xy. This gives you: (xy * 3) / (xy + x).
However, the xy's do not cancel out completely because of the addition in the denominator. The denominator, xy + x, can be rewritten as x(y + 1).
Now, you can cancel out the xy's, which leaves you with 3 / (y + 1) as the simplified expression. The xy terms don't cancel completely because the addition in the denominator prevents it.
3) Let's simplify the expression (4y - 8y^2) / (10y - 5) step by step:
Step 1: Factor out the common factor of 4y from the numerator: 4y(1 - 2y) / (10y - 5).
Step 2: Observe that the denominator is a binomial that can be factored further. The denominator, 10y - 5, can be factored as 5(2y - 1). So now the expression becomes: 4y(1 - 2y) / 5(2y - 1).
Step 3: Observe that there is a common factor of (1 - 2y) in both the numerator and the denominator. Cancel out this common factor: 4y * 1 / 5.
Step 4: Simplify the remaining expression: 4y / 5.
Therefore, the expression (4y - 8y^2) / (10y - 5) simplifies to -4y / 5.