Understanding Quadratic Polynomials with Given Roots: A Comprehensive Guide

Quadratic polynomials are a fundamental concept in algebra and are essential in solving a wide range of mathematical problems. A quadratic polynomial is a polynomial of degree two, generally written as ax^2 bx c 0. The roots of a quadratic polynomial are the solutions to the equation when set to zero.

The Given Problem and Its Solution

Consider a quadratic polynomial where one root is 5, and the sum of the roots is 0. Our task is to determine the quadratic polynomial and understand the process step-by-step.

Identifying the Roots

The problem states that one root is 5, and the sum of the roots is 0. If one root is 5, and the sum of the roots is 0, the other root must be -5. This is because if the roots are 5 and -5, their sum is indeed 0 (5 (-5) 0).

Formulating the Quadratic Polynomial

Given the roots of a quadratic polynomial, we can express it in factored form as (x - root1)(x - root2) 0. Using the roots we identified, the polynomial takes the form:

[(x - 5)(x - (-5)) (x - 5)(x 5) x^2 - 25.]

Thus, the quadratic polynomial is x^2 - 25. This polynomial can be verified by expanding the binomial product (x - 5)(x 5), which simplifies to x^2 - 25 (using the difference of squares formula, a^2 - b^2 (a - b)(a b)).

Further Explorations into Quadratic Polynomials

While solving the given problem, it's important to understand the broader context and implications of quadratic polynomials.

Form of a Quadratic Polynomial

A standard quadratic polynomial is written as ax^2 bx c 0. The coefficients a, b, and c are real numbers, and a ≠ 0. The roots of a quadratic polynomial can be found using the quadratic formula:

[text{Roots} frac{-b pm sqrt{b^2 - 4ac}}{2a}.]

The discriminant, given by D b^2 - 4ac, determines the nature of the roots. If D > 0, the polynomial has two distinct real roots; if D 0, the polynomial has one real root (a repeated root); and if D

The Sum of the Roots

In a quadratic polynomial, the sum of the roots can be directly related to the coefficients of the polynomial. According to Vieta's formulas, for a quadratic polynomial ax^2 bx c 0, the sum of the roots is given by:

[-frac{b}{a}.]

In the given problem, the sum of the roots is 0, which aligns with the values of 5 and -5. Therefore, a 1, b 0, and c -25, confirming the polynomial x^2 - 25.

Applications of Quadratic Polynomials

Quadratic polynomials have numerous applications in various fields, including physics, engineering, and economics. For instance, in physics, they are used to describe the motion of objects under constant acceleration. In economics, they can model the relationship between price and demand, where the vertex of a parabola (representing the quadratic polynomial) might indicate the maximum profit.

Conclusion

In conclusion, understanding quadratic polynomials and their roots is crucial for solving a variety of mathematical problems. The given problem illustrates the process of finding a quadratic polynomial given its roots and the sum of its roots. By grasping these fundamentals, one can approach more complex algebraic and real-world problems more effectively.

Key Takeaways

One root is 5, sum of the roots is 0 implies the other root is -5. The quadratic polynomial is x^2 - 25. The sum of the roots is given by -b/a.

Further Reading

For a deeper dive into the mathematics behind quadratic polynomials and their applications, consider exploring resources on algebraic theories and their applications in advanced mathematics, physics, and economics.