Finding Polynomials With Transformed Roots A Step By Step Guide

In mathematics, particularly in algebra, quadratic polynomials play a crucial role. These polynomials, expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0, have two roots or zeroes, which are the values of x for which f(x) = 0. Understanding the relationship between the roots of a quadratic polynomial and its coefficients is fundamental in solving various algebraic problems. This article delves into the process of finding a new polynomial whose roots are transformations of the roots of a given quadratic polynomial. Specifically, we will address the problem of finding a polynomial whose zeroes are 2α + 1 and 2β + 1, where α and β are the zeroes of the quadratic polynomial f(x) = x² + x - 2. This exploration will not only enhance your understanding of quadratic equations but also provide valuable techniques for manipulating polynomial roots.

Before diving into the solution, it's essential to grasp the basics of quadratic polynomials and their roots. A quadratic polynomial is a polynomial of degree two. The general form of a quadratic polynomial is f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The roots (or zeroes) of a quadratic polynomial are the values of x that make the polynomial equal to zero. These roots can be found using various methods, such as factoring, completing the square, or the quadratic formula.

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. Given a quadratic equation ax² + bx + c = 0, the roots are given by:

x = (-b ± √(b² - 4ac)) / (2a)

In addition to the quadratic formula, there are important relationships between the roots and the coefficients of a quadratic polynomial. If α and β are the roots of the quadratic polynomial f(x) = ax² + bx + c, then:

• Sum of the roots: α + β = -b/a
• Product of the roots: αβ = c/a

These relationships are crucial for solving problems involving transformations of roots, as they allow us to express the sum and product of the new roots in terms of the coefficients of the original polynomial. In the context of our problem, these relationships will help us find a new polynomial with roots 2α + 1 and 2β + 1.

The central problem we aim to solve is: Given the quadratic polynomial f(x) = x² + x - 2 with zeroes α and β, find a polynomial whose zeroes are 2α + 1 and 2β + 1. This problem involves transforming the roots of the original polynomial and constructing a new polynomial with these transformed roots. To tackle this, we will use the relationships between the roots and coefficients of a quadratic polynomial. Understanding these relationships is key to expressing the sum and product of the new roots in terms of the original roots α and β, which in turn can be related to the coefficients of f(x).

To solve the problem, we will follow a step-by-step approach that leverages the relationships between roots and coefficients of quadratic polynomials. This methodical approach will ensure clarity and accuracy in our solution.

Step 1: Identify the Coefficients of the Given Polynomial

First, we need to identify the coefficients of the given quadratic polynomial f(x) = x² + x - 2. Comparing this with the general form ax² + bx + c, we find that:

• a = 1
• b = 1
• c = -2

These coefficients will be used to find the sum and product of the roots α and β.

Step 2: Find the Sum and Product of the Roots α and β

Using the relationships between the roots and coefficients of a quadratic polynomial, we can find the sum and product of the roots α and β:

• Sum of the roots: α + β = -b/a = -1/1 = -1
• Product of the roots: αβ = c/a = -2/1 = -2

These values will be crucial in determining the sum and product of the transformed roots.

Step 3: Determine the New Roots

The new roots are given as 2α + 1 and 2β + 1. These are transformations of the original roots α and β.

Step 4: Calculate the Sum of the New Roots

Let's find the sum of the new roots:

(2α + 1) + (2β + 1) = 2α + 2β + 2 = 2(α + β) + 2

We know that α + β = -1, so we substitute this value:

2(-1) + 2 = -2 + 2 = 0

Therefore, the sum of the new roots is 0.

Step 5: Calculate the Product of the New Roots

Next, we calculate the product of the new roots:

(2α + 1)(2β + 1) = 4αβ + 2α + 2β + 1 = 4αβ + 2(α + β) + 1

We know that αβ = -2 and α + β = -1, so we substitute these values:

4(-2) + 2(-1) + 1 = -8 - 2 + 1 = -9

Thus, the product of the new roots is -9.

Step 6: Construct the New Polynomial

Now that we have the sum and product of the new roots, we can construct the new polynomial. If α' and β' are the new roots, then the new quadratic polynomial can be written in the form:

g(x) = x² - (α' + β')x + α'β'

Substituting the sum and product of the new roots, we get:

g(x) = x² - (0)x + (-9) = x² - 9

Therefore, the polynomial with zeroes 2α + 1 and 2β + 1 is g(x) = x² - 9.

An alternative method to solve this problem involves a direct transformation of the variable in the original polynomial. This method provides a different perspective and can be particularly useful in more complex scenarios.

Step 1: Express the Original Polynomial in Terms of the New Root

Let y be the new root, such that y = 2x + 1. We want to express x in terms of y:

y = 2x + 1 y - 1 = 2x x = (y - 1) / 2

Step 2: Substitute into the Original Polynomial

Now, substitute x = (y - 1) / 2 into the original polynomial f(x) = x² + x - 2:

f((y - 1) / 2) = ((y - 1) / 2)² + ((y - 1) / 2) - 2

Step 3: Simplify the Expression

Simplify the expression to obtain a polynomial in terms of y:

f((y - 1) / 2) = (y² - 2y + 1) / 4 + (y - 1) / 2 - 2

To eliminate fractions, multiply the entire equation by 4:

4f((y - 1) / 2) = y² - 2y + 1 + 2(y - 1) - 8

4f((y - 1) / 2) = y² - 2y + 1 + 2y - 2 - 8

4f((y - 1) / 2) = y² - 9

Step 4: Write the New Polynomial

Thus, the new polynomial in terms of y is y² - 9. Replacing y with x, we get the polynomial g(x) = x² - 9, which is the same result as before.

In this article, we successfully found a polynomial whose zeroes are transformations of the zeroes of a given quadratic polynomial. We started with the quadratic polynomial f(x) = x² + x - 2 and found a new polynomial with zeroes 2α + 1 and 2β + 1, where α and β are the zeroes of f(x). We used two methods:

1. Using the relationships between roots and coefficients: This method involved finding the sum and product of the original roots, then using these values to find the sum and product of the transformed roots, and finally constructing the new polynomial.
2. Direct transformation of the variable: This method involved expressing the original polynomial in terms of the new root by substituting x = (y - 1) / 2 into the original polynomial and simplifying the expression.

Both methods led to the same result: the polynomial with zeroes 2α + 1 and 2β + 1 is g(x) = x² - 9. This exercise demonstrates the power and versatility of algebraic techniques in manipulating polynomials and their roots. Understanding these concepts is crucial for further studies in mathematics, particularly in algebra and calculus. The ability to transform roots and construct new polynomials based on these transformations is a valuable skill in solving a wide range of mathematical problems. By mastering these techniques, you can approach similar problems with confidence and precision.

Quadratic Polynomials, Polynomial Roots, Root Transformation, Algebraic Techniques, Mathematics, Sum and Product of Roots

Finding Polynomials with Transformed Roots A Step by Step Guide

Original Question: यदि किसी द्विघात बहुपद $f(x)=x^2+x-2$ के शून्यक $\alpha$ तथा $\beta$ हैं, तब एक ऐसा बहुपद ज्ञात कीजिए जिसका शून्यक $(2 \alpha+1)$ और $(2 \beta+1)$ हैं।

Rewritten Questions for Clarity:

1. Given a quadratic polynomial f(x) = x² + x - 2 with roots α and β, how can we find a new polynomial whose roots are 2α + 1 and 2β + 1?
2. What is the process to determine a polynomial with roots 2α + 1 and 2β + 1, where α and β are the roots of the quadratic equation x² + x - 2 = 0?
3. If α and β are the zeroes of the quadratic polynomial f(x) = x² + x - 2, what steps are involved in constructing a polynomial with roots that are linear transformations of α and β, specifically 2α + 1 and 2β + 1?