Answer :
To find the resultant of two polynomials, denoted as [tex]\(\operatorname{r-C} \cdot D\)[/tex], we need to go through a few steps. The resultant of two polynomials is a scalar value that helps determine whether the polynomials have a common root.
Given the polynomials:
[tex]\[ P(x) = 6x^3 - 30x^2 + 60x - 48 \][/tex]
[tex]\[ Q(x) = 3x^3 - 12x^2 + 21x - 18 \][/tex]
we will compute the resultant of these two polynomials.
### Step-by-Step Solution:
1. Identify the coefficients of both polynomials:
For [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = 6x^3 - 30x^2 + 60x - 48
\][/tex]
Coefficients: [tex]\( [6, -30, 60, -48] \)[/tex]
For [tex]\( Q(x) \)[/tex]:
[tex]\[
Q(x) = 3x^3 - 12x^2 + 21x - 18
\][/tex]
Coefficients: [tex]\( [3, -12, 21, -18] \)[/tex]
2. Set up the Sylvester matrix:
The Sylvester matrix of two polynomials of degree [tex]\(n\)[/tex] and [tex]\(m\)[/tex] respectively, is an [tex]\((n + m) \times (n + m)\)[/tex] matrix composed from their coefficients.
For [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex], both are cubic polynomials, so the Sylvester matrix is a [tex]\(6 \times 6\)[/tex] matrix. The entries are the coefficients of [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].
3. Fill in the Sylvester matrix with the coefficients:
[tex]\[
S = \begin{bmatrix}
6 & -30 & 60 & -48 & 0 & 0 \\
0 & 6 & -30 & 60 & -48 & 0 \\
0 & 0 & 6 & -30 & 60 & -48 \\
3 & -12 & 21 & -18 & 0 & 0 \\
0 & 3 & -12 & 21 & -18 & 0 \\
0 & 0 & 3 & -12 & 21 & -18 \\
\end{bmatrix}
\][/tex]
4. Calculate the determinant of the Sylvester matrix.
The determinant of this matrix will give the resultant of [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].
5. Analyze the determinant result:
After calculating the determinant of the Sylvester matrix, we find that the result is 0.
Since the resultant of the two polynomials is 0, it indicates that [tex]\( 6x^3 - 30x^2 + 60x - 48 \)[/tex] and [tex]\( 3x^3 - 12x^2 + 21x - 18 \)[/tex] have a common root.
Therefore, the resultant (denoted as [tex]\(\operatorname{r-C} \cdot D\)[/tex]) of the given polynomials is:
[tex]\[
\boxed{0}
\][/tex]
Given the polynomials:
[tex]\[ P(x) = 6x^3 - 30x^2 + 60x - 48 \][/tex]
[tex]\[ Q(x) = 3x^3 - 12x^2 + 21x - 18 \][/tex]
we will compute the resultant of these two polynomials.
### Step-by-Step Solution:
1. Identify the coefficients of both polynomials:
For [tex]\( P(x) \)[/tex]:
[tex]\[
P(x) = 6x^3 - 30x^2 + 60x - 48
\][/tex]
Coefficients: [tex]\( [6, -30, 60, -48] \)[/tex]
For [tex]\( Q(x) \)[/tex]:
[tex]\[
Q(x) = 3x^3 - 12x^2 + 21x - 18
\][/tex]
Coefficients: [tex]\( [3, -12, 21, -18] \)[/tex]
2. Set up the Sylvester matrix:
The Sylvester matrix of two polynomials of degree [tex]\(n\)[/tex] and [tex]\(m\)[/tex] respectively, is an [tex]\((n + m) \times (n + m)\)[/tex] matrix composed from their coefficients.
For [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex], both are cubic polynomials, so the Sylvester matrix is a [tex]\(6 \times 6\)[/tex] matrix. The entries are the coefficients of [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].
3. Fill in the Sylvester matrix with the coefficients:
[tex]\[
S = \begin{bmatrix}
6 & -30 & 60 & -48 & 0 & 0 \\
0 & 6 & -30 & 60 & -48 & 0 \\
0 & 0 & 6 & -30 & 60 & -48 \\
3 & -12 & 21 & -18 & 0 & 0 \\
0 & 3 & -12 & 21 & -18 & 0 \\
0 & 0 & 3 & -12 & 21 & -18 \\
\end{bmatrix}
\][/tex]
4. Calculate the determinant of the Sylvester matrix.
The determinant of this matrix will give the resultant of [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex].
5. Analyze the determinant result:
After calculating the determinant of the Sylvester matrix, we find that the result is 0.
Since the resultant of the two polynomials is 0, it indicates that [tex]\( 6x^3 - 30x^2 + 60x - 48 \)[/tex] and [tex]\( 3x^3 - 12x^2 + 21x - 18 \)[/tex] have a common root.
Therefore, the resultant (denoted as [tex]\(\operatorname{r-C} \cdot D\)[/tex]) of the given polynomials is:
[tex]\[
\boxed{0}
\][/tex]
