What is the quotient in simplest form, and state the restrictions on the variable.
y^2 - 5y + 6 over y^3 PLUS y^2 + 3y - 10 over 4y^2
Can someone help?? I have ZERO clue how to do this.
I MEANT DIVIDED, NOT PLUS, SORRY
[(y^2-5y+6) / y^3] / [(y^2+3y-10) / (4y^2) ] ????
[(y-2)(y-3) / y^3] / [(y-2)(y+5) / (4y^2) ]
[(y-3) / y^3] / [(y+5) / (4y^2) ]
[(y-3) ] / [(y+5)y / (4) ]
[4(y-3) ] / [(y+5)y ]
Did you post this as "Sonsetto" earlier?
It was just as confusing then as it is now.
I am guessing you have something like this:
(y^2 - 5y + 6)/(y^3) + (y^2 + 3y - 10)/(4y^2)
or did you mean:
[ (y^2 - 5y + 6)/(y^3) ] / [ (y^2 + 3y - 10)/(4y^2) ]
those brackets are absolutely essential, waiting for your clarification.
I have a hunch that if the students were able to get the parentheses right they could do the problems :)
Of course! I'm here to help you understand and solve this problem step-by-step.
To simplify the quotient (division), we need to follow these steps:
Step 1: Factor both the numerator and the denominator.
Step 2: Cancel out any common factors between the numerator and the denominator.
Step 3: Write the resulting expression as the quotient in simplest form.
Step 1: Factoring the Numerator
For the numerator, we have y^2 - 5y + 6. To factorize this expression, we need to find two numbers that multiply to give 6 and add up to -5.
The factors of 6 are: 1, 2, 3, 6
After trying out different combinations, we find that -2 and -3 satisfy both conditions:
-2 * -3 = 6
-2 + (-3) = -5
Hence, the numerator can be factored to (y - 2)(y - 3).
Step 1: Factoring the Denominator
For the denominator, we have y^3 + y^2 + 3y - 10. To factorize this expression, we use the Rational Root Theorem and synthetic division. However, in this case, the polynomial does not have any rational roots. Therefore, we can assume it is fully factored.
The denominator cannot be simplified any further.
Step 2: Cancel Out Common Factors
Now, let's see if there are any common factors between the numerator and the denominator.
The numerator (y - 2)(y - 3) does not share any common factors with the denominator.
Step 3: Write the Quotient in Simplest Form
Since there are no common factors to cancel out, we can write the quotient in simplest form as:
(y^2 - 5y + 6) / (y^3 + y^2 + 3y - 10)
Please note that this can be considered the simplest form because the numerator is not factorable any further and the denominator is fully factored without any common factors.
Restrictions on the Variable:
To find the restrictions on the variable, we need to identify any values that would make the denominator equal to zero. In this case, the denominator is y^3 + y^2 + 3y - 10.
We can set the denominator equal to zero and solve for y:
y^3 + y^2 + 3y - 10 = 0
Unfortunately, this equation cannot be easily factored, and finding exact solutions might be challenging. You may need to use numerical methods (such as graphing or approximation techniques) to find the values of y that make the denominator equal to zero.
These values would be considered the restrictions on the variable y.