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Answer :
To find the solutions to the equation [tex]\(0 = x^3 - 19x^2 + 121x - 259\)[/tex], we will look for values of [tex]\(x\)[/tex] that satisfy this equation.
### Step 1: Understand the Equation
The given equation is a cubic polynomial:
[tex]\[ f(x) = x^3 - 19x^2 + 121x - 259 \][/tex]
Cubic equations typically have up to three roots (solutions). These roots can be real or complex.
### Step 2: Solve the Equation
For a cubic polynomial, finding roots might require methods such as factoring, using the Rational Root Theorem, or applying numerical methods. The goal is to find values of [tex]\(x\)[/tex] that make the equation equal to zero.
### Step 3: Identify the Solutions
The solutions to the polynomial equation [tex]\(x^3 - 19x^2 + 121x - 259 = 0\)[/tex] are:
1. [tex]\(x = 7\)[/tex]
2. [tex]\(x = 6 - i\)[/tex]
3. [tex]\(x = 6 + i\)[/tex]
### Explanation of the Solutions
1. Real Root:
- [tex]\(x = 7\)[/tex] is a real number and satisfies the equation.
2. Complex Roots:
- [tex]\(x = 6 - i\)[/tex] and [tex]\(x = 6 + i\)[/tex] are complex conjugates, reflecting the fact that the coefficients of the polynomial are real numbers. Complex roots often appear in conjugate pairs.
### Conclusion
The solutions to the equation [tex]\(x^3 - 19x^2 + 121x - 259 = 0\)[/tex] include one real root [tex]\(x = 7\)[/tex] and two complex roots [tex]\(6 - i\)[/tex] and [tex]\(6 + i\)[/tex]. These roots satisfy the original cubic equation.
### Step 1: Understand the Equation
The given equation is a cubic polynomial:
[tex]\[ f(x) = x^3 - 19x^2 + 121x - 259 \][/tex]
Cubic equations typically have up to three roots (solutions). These roots can be real or complex.
### Step 2: Solve the Equation
For a cubic polynomial, finding roots might require methods such as factoring, using the Rational Root Theorem, or applying numerical methods. The goal is to find values of [tex]\(x\)[/tex] that make the equation equal to zero.
### Step 3: Identify the Solutions
The solutions to the polynomial equation [tex]\(x^3 - 19x^2 + 121x - 259 = 0\)[/tex] are:
1. [tex]\(x = 7\)[/tex]
2. [tex]\(x = 6 - i\)[/tex]
3. [tex]\(x = 6 + i\)[/tex]
### Explanation of the Solutions
1. Real Root:
- [tex]\(x = 7\)[/tex] is a real number and satisfies the equation.
2. Complex Roots:
- [tex]\(x = 6 - i\)[/tex] and [tex]\(x = 6 + i\)[/tex] are complex conjugates, reflecting the fact that the coefficients of the polynomial are real numbers. Complex roots often appear in conjugate pairs.
### Conclusion
The solutions to the equation [tex]\(x^3 - 19x^2 + 121x - 259 = 0\)[/tex] include one real root [tex]\(x = 7\)[/tex] and two complex roots [tex]\(6 - i\)[/tex] and [tex]\(6 + i\)[/tex]. These roots satisfy the original cubic equation.
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