Constructing a Polynomial from Modified Roots Using Vieta’s Formulas

In algebra, we often need to find a polynomial whose roots are related to the roots of an existing polynomial. This can be particularly useful in various mathematical and engineering applications. For instance, if we know the roots of a cubic polynomial, we can construct a new polynomial whose roots are the products of these roots taken two at a time, and so on. In this article, we will demonstrate how to use Vieta’s formulas to find a polynomial with roots that are the products of the original roots of a given equation. Specifically, we will consider the equation (x^3 - 5x^2 - 3x 2 0) and find the polynomial whose roots are (ab, bc,) and (ac).

Understanding Vieta’s Formulas

Vietas formulas relate the coefficients of a polynomial to sums and products of its roots. Given a polynomial equation of the form:

[x^n a_{n-1}x^{n-1} cdots a_1x a_0 0]

if the roots of this polynomial are (a_1, a_2, ldots, a_n), then:

(a_1 a_2 cdots a_n -a_{n-1}) (a_1a_2 a_1a_3 cdots a_{n-1}a_n a_{n-2}) (a_1a_2a_3 cdots a_n (-1)^n a_0)

Applying Vieta’s Formulas to Given Polynomials

Given the polynomial (x^3 - 5x^2 - 3x 2 0), we have:

The sum of the roots is (a b c -frac{-5}{1} 5) The sum of the products of the roots taken two at a time is (ab ac bc frac{-3}{1} -3) The product of the roots is (abc -frac{2}{1} -2)

Constructing the New Polynomial

Let’s denote the new roots as (p ab, q bc, r ac). We aim to find a polynomial with these roots.

Step 1: Calculate the Sum of the New Roots

The sum of the new roots (p q r) can be expressed as:

(p q r ab bc ac 2abc 2(-2) -4)

Step 2: Calculate the Sum of the Products of the New Roots Taken Two at a Time

The sum of the products of the new roots taken two at a time is:

(pq qr rp ab^2 b^2c bc^2 a^2c ac^2 a^2b)

Collecting like terms, we get:

(pq qr rp 3ab 3ac 3bc 3(ab ac bc))

Since (ab ac bc -3), we have:

(pq qr rp 3(-3) -9)

Step 3: Calculate the Product of the New Roots

The product of the new roots (pqr) is:

(pqr (ab)(bc)(ac) a^2b^2c^2 (abc)^2 (-2)^2 4)

Forming the Polynomial

Using Vieta’s formulas, the polynomial with roots (p, q, r) is given by:

[x^3 - (p q r)x^2 (pq qr rp)x - pqr 0]

Substituting the values we found:

[x^3 - (-4)x^2 - 9x - 4 0]

Final Answer

The polynomial with roots (ab, bc, ac) is:

(x^3 4x^2 - 9x - 4 0)

Thus, the equation with roots (ab, bc, ac) is:

(boxed{x^3 4x^2 - 9x - 4 0})