factoring cubic polynomial equation using Cramer's rule.

Question

1) I have question about factoring cubic polynomials. In my note it says "Any polynomial equation with positive powers whose coefficients add to 0 will have a root of 1. Another, if sum of the coefficients of the even powers = sum of coefficient of the odd powers, -1 is a rood." However, all of my problems were solved with factor(x-1) if the coefficients add to 0 and vice versa. I am totally confused when to use 1, when to use -1. example:

x⁴+17x³-3x+k = 0

so the root is as follow:

a) 1 k = -49 b) -1 k = -12

2) 2nd questions is finding determinant 3X3 ,we repeat 1st and 2nd row and make 3 crisscross product the left and to the right. But i have no clue what is saying here. "where delta Dij = the (n-1)X(n-1) determinant former striking out row i and col j." example:

1 3 5 6 8
6 7 2 7 1
0-1 6 4 10
3 4 5 1 6 
8 7 9 4 -2

Dij = 6 7 2 1 
      0-1 6 10
      3 4 5 6
      8 7 9-2
for n = 3 what is value of Delta.

My questions is i do not understand which row and which column are referred in this. Any explaining would be highly appreciated. thanks.

Answer 1

If the root is 1 the polynomial is divisible by x -1; if the root is -1 the polynomial is divisible by x + 1. Unfortunately I'm getting k = -15 for the root of 1: adding the coefficients we have 1 + 17 -3 + k = 0 so 15 + k = 0. and k = -15. I checked it out by actually dividing through by x -1.

For the second the coefficients of the even powers add to 1 + k. The coefficients of the odd powers add to 17 - 3 = 14. So 1 + k = 14 and k = 13. Again, I double checked this with division.

The problem I see here is that you were not offered x = 1 k = 15 or x = -1 k = 13. Did you give us the wrong polynomial?

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Start with any matrix. Then D$_{ij}$ is the determinant of a submatrix that you get by crossing out the ith row and jth column of the original matrix. For example if A is a 3x3 matrix defined as

\begin{pmatrix} a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}

The D$_{23} is the determinant of

\begin{pmatrix} a & b \\ g & h \end{pmatrix}

which is ah-bg. To get the determinant of the 3x3 matrix above you can use $a*D_{11} - b*D_{12} + c*D_{13}$.