(2x . 3x + 1) divide by (2-x . 3x-1)
2 divide by 2x-1
To simplify the expression (2x * 3x + 1) divided by (2 - x * 3x - 1), we need to:
1. Expand the numerator:
(2x * 3x + 1) = 6x^2 + 1
2. Expand the denominator by applying the distributive property:
(2 - x * 3x - 1) = 2 - 3x^2 - x = -3x^2 - x + 2
Now, we can rewrite the expression as (6x^2 + 1) divided by (-3x^2 - x + 2).
To divide by 2x - 1, we need to perform polynomial long division:
1. Divide the first term of the numerator (6x^2) by the first term of the denominator (-3x^2). The result is (-2).
2. Multiply the entire denominator (-3x^2 - x + 2) by -2 and subtract it from the numerator (6x^2 + 1).
-2 * (-3x^2 - x + 2) = 6x^2 + 2x - 4
(6x^2 + 1) - (6x^2 + 2x - 4) = 2x + 5
3. Bring down the next term from the numerator (2x + 5).
4. Divide the first term of the new numerator (2x) by the first term of the denominator (-3x^2). The result is (-2/3x).
5. Multiply the entire denominator (-3x^2 - x + 2) by (-2/3x) and subtract it from the new numerator (2x + 5).
(-2/3x) * (-3x^2 - x + 2) = 2x + (2/3)
(2x + 5) - (2x + (2/3)) = (15/3) = 5
Since there are no more terms left to bring down or divide, the final result is -2 - (2/3x) + 5, which can be simplified as -2/3x + 3.
Therefore, the expression (2x * 3x + 1) divided by (2 - x * 3x - 1) is equal to -2/3x + 3.