Answer :
To find the zeros of the function [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex], we need to determine the values of [tex]\( x \)[/tex] where the function equals zero.
1. Identify the Function:
[tex]\[
f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6
\][/tex]
We are tasked with finding the zeros, which means solving the equation:
[tex]\[
x^4 - 19x^3 - 9x^2 + 19x - 6 = 0
\][/tex]
2. Approach to Solving:
- This polynomial is of degree 4, which is a quartic equation. Quartic equations can be solved using various methods, such as factoring (if possible), using the Rational Root Theorem, or applying numerical methods.
3. Finding Solutions:
- Given the nature of this polynomial, it is often complex and looks cumbersome to factor directly by hand. The polynomial may have real and/or complex roots.
- Advanced algebraic techniques or software can provide exact expressions or numerical approximations for these roots. Due to the complexity of quartic equations, their exact expressions are typically quite elaborate.
4. Zeros of the Polynomial:
- The zeros for this polynomial, determined through an appropriate solving method, are complex expressions. These can be expressed in terms of square roots and other operations resulting from solving the quartic.
In summary, the zeros of the polynomial [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex] are expressed as complex formulas involving square roots and other algebraic terms. These expressions provide the solutions to where [tex]\( f(x) \)[/tex] equals zero.
1. Identify the Function:
[tex]\[
f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6
\][/tex]
We are tasked with finding the zeros, which means solving the equation:
[tex]\[
x^4 - 19x^3 - 9x^2 + 19x - 6 = 0
\][/tex]
2. Approach to Solving:
- This polynomial is of degree 4, which is a quartic equation. Quartic equations can be solved using various methods, such as factoring (if possible), using the Rational Root Theorem, or applying numerical methods.
3. Finding Solutions:
- Given the nature of this polynomial, it is often complex and looks cumbersome to factor directly by hand. The polynomial may have real and/or complex roots.
- Advanced algebraic techniques or software can provide exact expressions or numerical approximations for these roots. Due to the complexity of quartic equations, their exact expressions are typically quite elaborate.
4. Zeros of the Polynomial:
- The zeros for this polynomial, determined through an appropriate solving method, are complex expressions. These can be expressed in terms of square roots and other operations resulting from solving the quartic.
In summary, the zeros of the polynomial [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex] are expressed as complex formulas involving square roots and other algebraic terms. These expressions provide the solutions to where [tex]\( f(x) \)[/tex] equals zero.
