Divide the polynomials. Express numbers using integers or simplified fractions. Check your answer by multiplication.
(6m^4n^5 - 6m^3n^4 - 2m^2 - 7) / (m^2n)
To divide the given polynomials, we will use the long division method.
6m^2n^4 - 6mn^3 - 2n - 7/m
We start by dividing the first term of the dividend, 6m^4n^5, by the divisor m^2n:
________
m^2n | 6m^4n^5 - 6m^3n^4 - 2m^2 - 7
6m^2n^4
Next, we multiply this quotient, 6m^2n^4, by the divisor m^2n, and subtract it from the dividend:
6m^4n^5 - 6m^3n^4 - 2m^2 - 7
- (6m^2n^4)
----------------------
- 6m^2n^5 + 6m^3n^4 - 2m^2 - 7
We repeat this process with the remaining terms:
___________________
m^2n | 6m^4n^5 - 6m^3n^4 - 2m^2 - 7
+ - 6m^2n^5
___________________
+ - 6m^2n^5 + 6m^3n^4
+ 6m^3n^4
___________________
+ 6m^3n^4 - 2m^2
+ 2m^2
___________________
+ 2m^2 - 7
+ 7
Now, we have fully divided the dividend by the divisor.
Therefore, the division expression is:
(6m^4n^5 - 6m^3n^4 - 2m^2 - 7) / (m^2n) = 6m^2n^4 + 6m^3n^3 + 2m^2 + (2m^2 - 7) / (m^2n)
To check our answer, we can multiply the divisor by our quotient:
(m^2n)(6m^2n^4 + 6m^3n^3 + 2m^2) + (m^2n)(2m^2 - 7)
= 6m^4n^5 + 6m^5n^4 + 2m^4n^2 + 2m^4n - 7m^2n
= 6m^4n^5 + 6m^5n^4 + 2m^4n^2 + 2m^4n - 7m^2n
Our answer is verified by multiplication.