Polynomial/Euclidean division/6x^3+x+1 over 3x^2+2x-4/Example

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We want to apply the Euclidean division (over )

So we want to divide a polynomial of degree by a polynomial of degree , hence the quotient and also the remainder have (at most) degree . For the first step, we ask with which term we have to multiply to achieve that the product and have the same leading term. This is . The product is

The difference between and this product is

We continue the division by with this polynomial, which we call . In order to get coincidence with the leading coefficient we have to multiply with . This yields

The difference between this and is therefore

This is the remainder and altogether we get