Answer:
13y^2+5y-1
Explanation:
In long division, we divide the dividend (the polynomial being divided) by the divisor (the polynomial we are dividing by) to get the quotient. The general steps are as follows:
1. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient. Write this term above the line.
2. Multiply the divisor by this first term of the quotient, and write the result underneath the first part of the dividend.
3. Subtract this result from the first part of the dividend, and write the remainder underneath.
4. Bring down the next term of the dividend to the right of the remainder.
5. Repeat steps 1-4 until there are no more terms in the dividend to bring down, or until the degree of the remainder is less than the degree of the divisor.
In your example, the dividend is 65y^3+28y^2-22y-5 and the divisor is 5y+1. We start by dividing the first term of the dividend, 65y^3, by the first term of the divisor, 5y, to get 13y^2. We write this term above the line as the first term of the quotient.
Next, we multiply the divisor by this first term of the quotient to get (5y+1)(13y^2) = 65y^3+13y^2. We write this underneath the first part of the dividend, and subtract it from the dividend to get:
13y^2
----------
5y + 1 | 65y^3 + 28y^2 - 22y - 5
-65y^3 - 13y^2
--------------
15y^2 - 22y
The result of this subtraction is 15y^2-22y. We write this as the remainder underneath the line, and bring down the next term of the dividend, which is -5. We now repeat the process of dividing the first term of the remainder by the first term of the divisor:
13y^2 + 5y
----------------
5y + 1 | 65y^3 + 28y^2 - 22y - 5
-65y^3 - 13y^2
--------------
15y^2 - 22y
15y^2 + 3y
------------
-25y - 5
We get 5y as the next term of the quotient, since 5y divided by 5y is 1. We multiply the divisor by this term to get (5y+1)(5y) = 25y^2+5y, and subtract this from the remainder to get -25y-5. We bring down the next term of the dividend, which is 0, and we see that we have no more terms to bring down. Therefore, our final quotient is 13y^2+5y-1, and our final remainder is 0.
Hope this helps!