Constructing a Polynomial Using Given Roots and Transformations
In the realm of polynomial equations, understanding how to manipulate and use given roots to construct new polynomials is a fundamental skill. This article will guide you through the process of creating a new polynomial based on specified transformations of the roots of a given quadratic equation. We will break down the steps and use detailed calculations to achieve our goal.
Given Polynomial and Roots
Consider a quadratic polynomial x^2 - 8x 6, which has roots α and β. Our task is to find the polynomial whose roots are 1 β/α and 1 α/β. This problem requires a systematic approach involving the manipulation of the given roots and the application of polynomial transformation techniques.
Step 1: Sum and Product of Given Roots
First, we use the sum and product of the roots formula for a quadratic equation ax^2 bx c 0 which are sum of roots -b/a and product of roots c/a. For the given equation:
sum of roots (α β) 8/1 8 product of roots (αβ) 6/1 6These values will be used to express the new roots in terms of the original roots.
Step 2: Expressing New Roots in Terms of α and β
The given new roots are 1 β/α and 1 α/β. We can rewrite them as:
1 β/α (α β)/α 8/α 1 α/β (α β)/β 8/βStep 3: Forming the New Polynomial
To form a polynomial with these new roots, we use the fact that if a polynomial has roots r1, r2, then it can be represented as:
(x - r1)(x - r2) 0 (x - 8/α)(x - 8/β) 0Expanding this expression, we get:
x^2 - (8/α 8/β)x (8/α)(8/β) 0Substituting the sum and product of roots from step 1, we get:
x^2 - (8)(-6/8)x (64/8) 0 x^2 6x 8 0Multiplying the entire equation by 3 to simplify, we obtain:
3x^2 - 32x 32 0Conclusion
The polynomial with roots 1 β/α and 1 α/β is 3x^2 - 32x 32. This process involves a combination of algebraic manipulation, polynomial properties, and a systematic approach to transforming the given roots.