Answer :
Certainly! 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] that make the function equal to zero. Let's go through the solution step-by-step:
1. Understanding the Problem:
We are given a polynomial of degree 4, [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex]. The roots or zeros of the polynomial are the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 0 \)[/tex].
2. Approach:
Finding the roots of a quartic polynomial (degree 4) can be complex, and typically requires specialized methods. Exact roots might not be integers or simple fractions, and could involve complex numbers or irrational numbers.
3. Analysis:
The solution involves using mathematical techniques that solve polynomial equations analytically. This can be done through factorization (if possible), application of the Rational Root Theorem, synthetic division, or more advanced algebraic methods for quartic equations.
4. Result:
Through this analytical process, we arrive at the roots:
[tex]\[
x_1 = \frac{19}{4} - \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3\,c} + 2c} + \frac{1}{2}\sqrt{\frac{385}{2} - 2c - \frac{7391}{4u} - \frac{182}{3c}}
\][/tex]
[tex]\[
x_2 = \frac{19}{4} + \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3c} + 2c} + \frac{1}{2}\sqrt{\frac{385}{2} - 2c + \frac{7391}{4u} - \frac{182}{3c}}
\][/tex]
[tex]\[
x_3 = \frac{19}{4} - \frac{1}{2}\sqrt{\frac{385}{2} - 2c - \frac{7391}{4u} - \frac{182}{3c}} - \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3c} + 2c}
\][/tex]
[tex]\[
x_4 = \frac{19}{4} - \frac{1}{2}\sqrt{\frac{385}{2} - 2c + \frac{7391}{4u} - \frac{182}{3c}} + \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3c} + 2c}
\][/tex]
Where:
- [tex]\( c = (-1543/8 + \sqrt{48163137}/72)^{1/3} \)[/tex]
- [tex]\( u = \sqrt{\frac{385}{4} + \frac{182}{3c} + 2c} \)[/tex]
5. Conclusion:
These expressions are the zeros of the polynomial function [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex]. They represent the points where the function crosses the x-axis (i.e., the function equals zero).
Finding such roots highlights the complexity of solving higher-degree polynomials and often requires symbolic computation. If your interest is in numerical solutions or graphical insights, computational tools or approximations are commonly used.
1. Understanding the Problem:
We are given a polynomial of degree 4, [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex]. The roots or zeros of the polynomial are the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 0 \)[/tex].
2. Approach:
Finding the roots of a quartic polynomial (degree 4) can be complex, and typically requires specialized methods. Exact roots might not be integers or simple fractions, and could involve complex numbers or irrational numbers.
3. Analysis:
The solution involves using mathematical techniques that solve polynomial equations analytically. This can be done through factorization (if possible), application of the Rational Root Theorem, synthetic division, or more advanced algebraic methods for quartic equations.
4. Result:
Through this analytical process, we arrive at the roots:
[tex]\[
x_1 = \frac{19}{4} - \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3\,c} + 2c} + \frac{1}{2}\sqrt{\frac{385}{2} - 2c - \frac{7391}{4u} - \frac{182}{3c}}
\][/tex]
[tex]\[
x_2 = \frac{19}{4} + \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3c} + 2c} + \frac{1}{2}\sqrt{\frac{385}{2} - 2c + \frac{7391}{4u} - \frac{182}{3c}}
\][/tex]
[tex]\[
x_3 = \frac{19}{4} - \frac{1}{2}\sqrt{\frac{385}{2} - 2c - \frac{7391}{4u} - \frac{182}{3c}} - \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3c} + 2c}
\][/tex]
[tex]\[
x_4 = \frac{19}{4} - \frac{1}{2}\sqrt{\frac{385}{2} - 2c + \frac{7391}{4u} - \frac{182}{3c}} + \frac{1}{2}\sqrt{\frac{385}{4} + \frac{182}{3c} + 2c}
\][/tex]
Where:
- [tex]\( c = (-1543/8 + \sqrt{48163137}/72)^{1/3} \)[/tex]
- [tex]\( u = \sqrt{\frac{385}{4} + \frac{182}{3c} + 2c} \)[/tex]
5. Conclusion:
These expressions are the zeros of the polynomial function [tex]\( f(x) = x^4 - 19x^3 - 9x^2 + 19x - 6 \)[/tex]. They represent the points where the function crosses the x-axis (i.e., the function equals zero).
Finding such roots highlights the complexity of solving higher-degree polynomials and often requires symbolic computation. If your interest is in numerical solutions or graphical insights, computational tools or approximations are commonly used.
