Answer :
To solve the polynomial equation [tex]\(x^4 + x^3 - 19x^2 + x - 20 = 0\)[/tex], we look for its roots, which are the values of [tex]\(x\)[/tex] that satisfy this equation.
Here are the steps to find the roots:
1. Understand the Polynomial: The equation is a quartic (fourth degree) polynomial, which means it can have up to four roots.
2. Identify Real Roots: After solving the equation, we find the real roots:
- [tex]\(x = -5\)[/tex]
- [tex]\(x = 4\)[/tex]
These are real numbers that make the polynomial equal to zero.
3. Identify Complex Roots: The polynomial also has complex roots, which are:
- [tex]\(x = -i\)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit, [tex]\(\sqrt{-1}\)[/tex])
- [tex]\(x = i\)[/tex]
These complex numbers also satisfy the equation when evaluated in the polynomial.
So, the complete set of roots for the polynomial equation [tex]\(x^4 + x^3 - 19x^2 + x - 20 = 0\)[/tex] is:
- [tex]\(x = -5\)[/tex]
- [tex]\(x = 4\)[/tex]
- [tex]\(x = -i\)[/tex]
- [tex]\(x = i\)[/tex]
These roots include both the real and the complex solutions to the equation.
Here are the steps to find the roots:
1. Understand the Polynomial: The equation is a quartic (fourth degree) polynomial, which means it can have up to four roots.
2. Identify Real Roots: After solving the equation, we find the real roots:
- [tex]\(x = -5\)[/tex]
- [tex]\(x = 4\)[/tex]
These are real numbers that make the polynomial equal to zero.
3. Identify Complex Roots: The polynomial also has complex roots, which are:
- [tex]\(x = -i\)[/tex] (where [tex]\(i\)[/tex] is the imaginary unit, [tex]\(\sqrt{-1}\)[/tex])
- [tex]\(x = i\)[/tex]
These complex numbers also satisfy the equation when evaluated in the polynomial.
So, the complete set of roots for the polynomial equation [tex]\(x^4 + x^3 - 19x^2 + x - 20 = 0\)[/tex] is:
- [tex]\(x = -5\)[/tex]
- [tex]\(x = 4\)[/tex]
- [tex]\(x = -i\)[/tex]
- [tex]\(x = i\)[/tex]
These roots include both the real and the complex solutions to the equation.
