Determining the Coefficients of a Quadratic Polynomial from Given Zeros

In this article, we will explore the process of finding the coefficients of a quadratic polynomial when the zeros of the polynomial are given. Specifically, we will focus on the example where the zeros of the quadratic polynomial (x^2 a_1x b) are 2 and -3. We will go through step-by-step derivations and explanations to determine the values of (a_1) and (b).

Understanding the Concept

A quadratic polynomial is a polynomial of degree 2, and it can be written in the standard form as (ax^2 bx c), where (a eq 0). The zeros of the polynomial are the roots of the equation (ax^2 bx c 0).

Given Zeros and Polynomial Form

Given that the zeros of the quadratic polynomial (x^2 a_1x b) are 2 and -3, we can express the polynomial in its factored form:

[x^2 a_1x b (x - 2)(x 3)]

Expanding the Factored Form

To find the coefficients (a_1) and (b), we need to expand the factored form and compare it with the standard form of the polynomial.

[x^2 a_1x b (x - 2)(x 3)]

Expanding the right-hand side:

[x^2 a_1x b x^2 3x - 2x - 6 x^2 x - 6]

Comparing Coefficients

By comparing the coefficients of the expanded form with the standard form (x^2 a_1x b), we can determine the values of (a_1) and (b).

[x^2 a_1x b x^2 x - 6]

Thus, we have:

[a_1 1]

[b -6]

Alternative Method Using Sum and Product of Roots

We can also use the properties of the sum and product of the roots of a quadratic polynomial to determine the coefficients.

Sum of Roots

The sum of the roots of the polynomial (ax^2 bx c 0) is given by (-frac{b}{a}). For the given polynomial (x^2 a_1x b), the sum of the roots is:

[2 (-3) -a_1]

[-1 -a_1]

[a_1 1]

Product of Roots

The product of the roots of the polynomial (ax^2 bx c 0) is given by (frac{c}{a}). For the given polynomial (x^2 a_1x b), the product of the roots is:

[2 cdot (-3) frac{b}{1}]

[-6 b]

Verification

Let's verify the solution by substituting the values of (a_1) and (b) into the polynomial and substituting the zeros back into the equation.

The polynomial is:

[x^2 x - 6]

Substituting (x 2):

[2^2 2 - 6 0]

[4 2 - 6 0]

Substituting (x -3):

[(-3)^2 (-3) - 6 0]

[9 - 3 - 6 0]

Both checks confirm that the polynomial (x^2 x - 6) has zeros at (x 2) and (x -3).

Conclusion

We determined that the coefficients of the quadratic polynomial (x^2 a_1x b) with zeros 2 and -3 are (a_1 1) and (b -6).

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