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).
If you're interested in more such problems or if you have any doubts, please follow this page and upvote it!