Answer :
To find all the zeros of the function and write its linear factorization, let's follow these steps:
1. Identify the Function:
We are given the polynomial function:
[tex]\( f(x) = 6x^4 - 23x^3 - 3x^2 + 321x - 1001 \)[/tex].
2. Find the Zeros:
The zeros of the function are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. Based on calculations (not shown here), the zeros of this polynomial are:
[tex]\[
x = -\frac{11}{3}, \quad x = \frac{7}{2}, \quad x = 2 - 3i, \quad x = 2 + 3i
\][/tex]
3. Write the Linear Factorization:
Once we have the zeros, we can write the function as a product of its linear factors. Each zero corresponds to a factor of the form [tex]\( (x - \text{zero}) \)[/tex].
- For [tex]\( x = -\frac{11}{3} \)[/tex], the factor is [tex]\( (x + \frac{11}{3}) \)[/tex].
- For [tex]\( x = \frac{7}{2} \)[/tex], the factor is [tex]\( (x - \frac{7}{2}) \)[/tex].
- For the complex zeros [tex]\( x = 2 - 3i \)[/tex] and [tex]\( x = 2 + 3i \)[/tex], the factors are [tex]\( (x - (2 - 3i)) \)[/tex] and [tex]\( (x - (2 + 3i)) \)[/tex] respectively.
Therefore, the linear factorization of the function is:
[tex]\[
f(x) = 6(x + \frac{11}{3})(x - \frac{7}{2})(x - (2 - 3i))(x - (2 + 3i))
\][/tex]
This linear factorization expresses the polynomial as a product of its linear factors directly related to its zeros.
1. Identify the Function:
We are given the polynomial function:
[tex]\( f(x) = 6x^4 - 23x^3 - 3x^2 + 321x - 1001 \)[/tex].
2. Find the Zeros:
The zeros of the function are the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 0 \)[/tex]. Based on calculations (not shown here), the zeros of this polynomial are:
[tex]\[
x = -\frac{11}{3}, \quad x = \frac{7}{2}, \quad x = 2 - 3i, \quad x = 2 + 3i
\][/tex]
3. Write the Linear Factorization:
Once we have the zeros, we can write the function as a product of its linear factors. Each zero corresponds to a factor of the form [tex]\( (x - \text{zero}) \)[/tex].
- For [tex]\( x = -\frac{11}{3} \)[/tex], the factor is [tex]\( (x + \frac{11}{3}) \)[/tex].
- For [tex]\( x = \frac{7}{2} \)[/tex], the factor is [tex]\( (x - \frac{7}{2}) \)[/tex].
- For the complex zeros [tex]\( x = 2 - 3i \)[/tex] and [tex]\( x = 2 + 3i \)[/tex], the factors are [tex]\( (x - (2 - 3i)) \)[/tex] and [tex]\( (x - (2 + 3i)) \)[/tex] respectively.
Therefore, the linear factorization of the function is:
[tex]\[
f(x) = 6(x + \frac{11}{3})(x - \frac{7}{2})(x - (2 - 3i))(x - (2 + 3i))
\][/tex]
This linear factorization expresses the polynomial as a product of its linear factors directly related to its zeros.
