Divide the polynomials. Express numbers using integers or simplified fractions. Check your answer by multiplication.
(8a^4b^5-6a^3b^4+4a^2b^3 - 1)/(a^2b)
To divide the polynomial (8a^4b^5-6a^3b^4+4a^2b^3 - 1) by (a^2b), we need to divide each term of the polynomial by (a^2b).
First, let's divide 8a^4b^5 by a^2b. Dividing the coefficients, we get 8 / 1 = 8. Subtracting the exponents in the variables, we get a^4 - a^2 = a^(4-2) = a^2. Dividing the b terms, we subtract the exponents: b^5 - b^1 = b^(5-1) = b^4. So, the first term becomes 8a^2b^4.
Now let's divide -6a^3b^4 by a^2b. Dividing the coefficients, we get -6 / 1 = -6. Subtracting the exponents in the variables, we get a^3 - a^2 = a^(3-2) = a. Dividing the b terms, we subtract the exponents: b^4 - b^1 = b^(4-1) = b^3. So, the second term becomes -6ab^3.
Next, let's divide 4a^2b^3 by a^2b. Dividing the coefficients, we get 4 / 1 = 4. Subtracting the exponents in the variables, we get a^2 - a^2 = a^(2-2) = a^0 = 1. Dividing the b terms, we subtract the exponents: b^3 - b^1 = b^(3-1) = b^2. So, the third term becomes 4b^2.
Lastly, let's divide -1 by a^2b. Dividing the coefficients, we get -1 / 1 = -1. Subtracting the exponents in the variables, we get 0 - a^2 = -a^2. Dividing the b terms, we get b^0 - b^1 = 1 - b = -b. So, the last term becomes -1.
Therefore, the division of the polynomial (8a^4b^5-6a^3b^4+4a^2b^3 - 1) by (a^2b) is:
8a^2b^4 - 6ab^3 + 4b^2 - 1
To check our answer, let's multiply this result by (a^2b):
(a^2b)(8a^2b^4 - 6ab^3 + 4b^2 - 1)
= 8a^4b^5 - 6a^3b^4 + 4a^2b^3 - a^2b - 8a^2b^4 + 6ab^3 - 4b^2 + b
= 8a^4b^5 - 6a^3b^4 + 4a^2b^3 - 8a^2b^4 + 6ab^3 - a^2b + b - 4b^2
= 8a^4b^5 - 14a^3b^4 + 10a^2b^3 - a^2b + b - 4b^2
This is equal to the original polynomial (8a^4b^5-6a^3b^4+4a^2b^3 - 1). Therefore, our answer is correct.