Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$

In summary, a polynomial is an algebraic expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The roots of a polynomial are the values of the variable that make the polynomial equal to zero. The most common way to find the roots of a polynomial is by factoring or using the quadratic formula. The expression "1/A + 1/B + 1/C" represents the sum of the reciprocals of three numbers, and its significance in relation to the roots of a polynomial may vary.
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anemone
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Let $p,\,q$ and $r$ be the distinct roots of the polynomial $x^3-22x^2+80x-67$. It is given that there exist real numbers $A,\,B$ and $C$ such that

$\dfrac{1}{s^3-22s^2+80s-67}=\dfrac{A}{s-p}+\dfrac{B}{s-q}+\dfrac{C}{s-r}$ for all $s\not \in \{p,\,q,\,r\}$. What is $\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{C}$?
 
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Is answer 244?
 

1. What are the roots of a polynomial?

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. They are also known as the solutions or zeroes of the polynomial.

2. How do you find the roots of a polynomial?

To find the roots of a polynomial, you can use various methods such as factoring, the quadratic formula, or synthetic division. The method used depends on the degree and complexity of the polynomial.

3. What is the significance of finding the roots of a polynomial?

Finding the roots of a polynomial is important because it helps us understand the behavior and properties of the polynomial. It also allows us to solve equations and make predictions in various fields such as physics, engineering, and economics.

4. What is the relationship between the roots and the coefficients of a polynomial?

The relationship between the roots and coefficients of a polynomial is given by Vieta's formulas. These formulas state that the sum of the roots is equal to the opposite of the coefficient of the second highest degree term, and the product of the roots is equal to the constant term divided by the coefficient of the highest degree term.

5. How do you find the sum of the reciprocals of the roots of a polynomial?

To find the sum of the reciprocals of the roots of a polynomial, you can use the formula:
1/A + 1/B + 1/C = (A+B+C)/(ABC)
where A, B, and C are the roots of the polynomial. This formula can be derived from Vieta's formulas.

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