Perform the indicated operations and simplify. 3z^2 - z - 2/6z^2 - 11z - 10 times 4z^2 - 25/2z^2 + 11z + 15 After I factored I got (3z + 2)(z - 1)/(3z + 2)(2z - 5) times (2z - 5)(2z + 5)/(2z - 5)(z + 3) After canceling I got (z - 1)(2z + 5)/(2z - 5)(z + 3). Where did I go wrong?
(3z^2-z-2)/(6z^2-11z-10)(4z^2-25)/(2z^2+11z+15)
(3z+2)(z-1)(2z-5)(2z+5)
----------------------------------
(3z+2)(2z-5)(2z+5)(z+3)
(z-1)/(z+3)
You had a typo: it was (2z+5)(z+3)
To find where you went wrong, let's go through the steps of factoring and simplifying the expression step by step:
Given expression: (3z^2 - z - 2) / (6z^2 - 11z - 10) * (4z^2 - (25/2)z^2 + 11z + 15)
Step 1: Factor the numerator and denominator separately.
For the first fraction, factor the numerator, which is 3z^2 - z - 2, and the denominator, which is 6z^2 - 11z - 10.
For the second fraction, factor the numerator, which is 4z^2 - (25/2)z^2 + 11z + 15, and the denominator, which is 1.
Factored expression: [(3z + 2)(z - 1)] / [(3z + 2)(2z - 5)] * [(2z - 5)(2z + 5)] / 1
Step 2: Simplify by canceling out common factors.
Notice that the expression (3z + 2) cancels out in the numerator and denominator. However, you mistakenly canceled out the expression (2z - 5) entirely, which is incorrect.
Correctly simplified expression: (z - 1)(2z + 5) / (3z + 2)(2z - 5)(2z + 5)
So, the mistake you made was canceling out the entire expression (2z - 5), which should have remained in the denominator.
It seems like you made a mistake when canceling out the common factors. Let's go through the simplification process step by step to identify the error.
First, let's look at the expression you started with:
(3z^2 - z - 2) / (6z^2 - 11z - 10) * (4z^2 - (25/2)z^2 + 11z + 15)
Now, let's factor each of the two quadratic expressions individually:
The first quadratic, 6z^2 - 11z - 10, can be factored as (3z + 2)(2z - 5).
The second quadratic, 4z^2 - (25/2)z^2 + 11z + 15, can be simplified to (2z^2 + 11z + 15).
Now, the expression becomes:
(3z^2 - z - 2) / (3z + 2)(2z - 5) * (2z^2 + 11z + 15)
Next, let's check if there are any common factors we can cancel from the numerator and denominator:
In the numerator, we have (3z^2 - z - 2), which cannot be factored further.
In the denominator, we have (3z + 2)(2z - 5), which cannot be simplified either.
Therefore, we cannot cancel any common factors. So the correct simplified expression would be:
(3z^2 - z - 2)(2z^2 + 11z + 15) / (3z + 2)(2z - 5)
Hope this clarifies your doubt. Let me know if you have any further questions!