1

I know how to calculate the resultant of two polynomials.. but I am little confused with an example I have come across in a past paper...

The type of example I am used to are f= $5X^3-185X-420$ and g=$-2X^3+218X-840$

I know the initial matrix would be

$\begin{bmatrix} 5 & 0 & 0 & -2 & 0 & 0 \\0 & 5 & 0 & 0 &-2 & 0 \\-185 & 0 &5&218&0&-2\\-420&-185&0&-840&218&0\\0&-420&-185&0&-840&218\\0&0&-420&0&0&-840 \end{bmatrix}$

But then I have an example where the two have different largest powers

f=$2X^3-8X^2-40X+96$ and g=$3X^2-17X-6$

Would the first matrix be...

$\begin{bmatrix} 2&0&0&0&0&0 \\ -8&2&0&3&0&0\\-40&-8&2&-17&3&0\\96&-40&-8&-6&-17&3\\0&96&-40&0&-6&-17\\0&0&96&0&0&-6 \end{bmatrix}$

Im very confused by this and the way in which the matrix should look for g

Any help wouldbe greatly appreciated as I have the exam tomorrow

Mathsstudent147
  • 1,495

1 Answers1

2

No; the size of the Sylvester matrix is the sum of the degrees. In your second case, you should have $$\begin{pmatrix} 2 & 0 & 3 & 0 & 0 \\ -8 & 2 & -17 & 3 & 0 \\ -40 & -8 & -6 & -17 & 3 \\ 96 & -40 & 0 & -6 & -17 \\ 0 & 96 & 0 & 0 & -6 \end{pmatrix}$$ that is, write the degree $m$ polynomial $n$ times, the degree $n$ polynomial $m$ times.

Cocopuffs
  • 10,307