Problem 29 - Calculus Single Variable

Question

Let \(r\) and \(s\) be the roots of \(x^{2}+A x+B\). Express the coefficients \(A\) and \(B\) as functions of \(r\) and \(s\).

Step-by-Step Solution

Answer

The coefficients are \(A = -(r + s)\) and \(B = rs\).

Step 1: Recall Vieta's Formulas

For a quadratic equation of the form \(x^2 + Ax + B = 0\), Vieta's formulas tell us that the sum of the roots \(r + s\) is equal to the negative of the coefficient of \(x\) divided by the leading coefficient, which is \(A\), and the product of the roots \(r \cdot s\) is equal to the constant term \(B\) divided by the leading coefficient. Since the leading coefficient here is 1, this simplifies to: \[ r + s = -A \] \[ r \cdot s = B \].

Step 2: Express Coefficient A in Terms of the Roots

From Vieta's formula, the sum of the roots is \(r + s = -A\). Solve for \(A\) by multiplying both sides by -1 to isolate \(A\): \[ A = -(r + s) \].

Step 3: Express Coefficient B in Terms of the Roots

Again using Vieta's formulas, the product of the roots is given by \(r \cdot s = B\). Therefore, the expression for \(B\) is simply: \[ B = rs \].

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.
One of the most important aspects of quadratic equations is their ability to be rewritten or factored in terms of their roots, which are typically represented as \( r \) and \( s \) for a standard quadratic equation.

• When \( a = 1 \) (as in the equation \( x^2 + Ax + B = 0 \)), it simplifies the process of finding the roots and allows us to directly use Vieta's formulas.
• The classic form shall be \( x^2 + Ax + B \) and simplifies the understanding and solving of these equations.

Understanding quadratic equations is pivotal as they often pop up in diverse areas of mathematics and real-world problems, from physics to finance. They are not just equations; they form parabola graphs, making them visually insightful.With the quadratic toolset, you can identify patterns or predict future system behaviors, which is powerful anytime you're dealing with anything that manifests as a curve or relationship that simulates quadratic characteristics.

Roots of a Polynomial

The roots of a polynomial are the solutions to the equation when the polynomial is set to zero.
In a quadratic equation like \( x^2 + Ax + B = 0 \), the roots \( r \) and \( s \) are the values of \( x \) that make this equation true.

• These roots represent the x-intercepts of the polynomial graphically.
• The sum and product of these roots are critical and are directly associated with the polynomial's coefficients.

For example, Vieta's formulas establish that for a polynomial of this kind, the sum of the roots \( r + s \) equals \(-A\) and their product \( r \cdot s \) equals \( B \). These are relational pointers that bridge the roots and the coefficients seamlessly. Understanding the roots and how they connect to the polynomial's shape and position is fundamental.They tell a lot about the nature and characteristics of the equation just by their values, giving us insights into the equation's behavior without graphing.

Coefficients in Terms of Roots

Coefficients in terms of roots can be explored through a fascinating mathematical shortcut provided by Vieta's formulas.
Essentially, instead of calculating coefficients from scratch, you can reverse-engineer them from the known roots.

• The coefficient \( A \) is simply found by negating the sum of the roots: \( A = -(r + s) \).
• Similarly, the coefficient \( B \) comes directly from the product of the roots: \( B = r \cdot s \).

As illustrated, these formulas offer us a way to express polynomial equations purely in terms of their roots.This relationship is vital, particularly in cases where the roots are derived or measured quantities, and coefficients need to be inferred accurately for mathematical models.Understanding these relationships isn't just about solving equations; it involves manipulating the equations respectfully to unveil deeper insights into their structures and meanings behind them.