Forming a Quadratic Polynomial Given the Sum and Product of Its Zeros
When dealing with polynomial equations, it's sometimes necessary to form a quadratic polynomial given the sum and product of its zeros. This article explains the process step-by-step and provides a detailed solution process for the specific case where the sum of the zeros is 3 and the product of the zeros is -2/5.
The Standard Form of a Quadratic Polynomial
A quadratic polynomial in standard form can be written as:
px2 - (sum of zeros)x - (product of zeros) 0
Using this formula, if we are given the sum and product of the zeros, we can construct the desired quadratic polynomial. Let's apply this method to the given example: the sum of the zeros is 3 and the product of the zeros is -2/5.
Step-by-Step Solution
Step 1: Write down the given values.
Sum of zeros (S) 3
Product of zeros (P) -2/5
Step 2: Apply the formula.
The quadratic polynomial can be formed as:
x2 - 3x - (-2/5) 0
The negative sign in the product term is changed to positive for simplicity:
x2 - 3x 2/5 0
Step 3: Eliminate the fraction.
To get rid of the fraction, we can multiply the entire equation by 5:
5x2 - 15x 2 0
However, since the product of the zeros is -2/5, we should adjust the equation accordingly:
5x2 - 15x - 2 0
Final Quadratic Polynomial
The quadratic polynomial with the sum of zeros 3 and the product of zeros -2/5 is:
5x2 - 15x - 2 0
Conclusion
Forming a quadratic polynomial given the sum and product of its zeros is a straightforward process when you follow the standard quadratic form. By utilizing the sum and product of the zeros, you can easily write down the polynomial equation. In this specific example, we obtained the polynomial:
5x2 - 15x - 2 0
Additional Notes
Remember that the coefficient of x2 (k) can be any non-zero number. In our example, we used 5 to eliminate the fractional coefficient. The equation can be written in different forms, but the core steps remain the same.
Understanding these steps will help you solve similar problems involving quadratic polynomials quickly and accurately. If you have any further questions, feel free to ask!