if alpha ,beta are the zeroes of a polynomial,such that alpha+beta=6 and alpha into beta=4 ,then write the polynomial.
To find the polynomial, we need to use the relationship between the zeros of a polynomial and its coefficients.
First, let's denote the polynomial as P(x). Since α and β are the zeroes of the polynomial, we know that:
P(α) = 0 and P(β) = 0
We are given that α + β = 6 and α * β = 4. We can use these values to form equations that relate to the coefficients of the polynomial.
The sum of the roots states that:
α + β = -b/a (Equation 1)
And the product of the roots states that:
α * β = c/a (Equation 2)
Here, a, b, and c are the coefficients of the polynomial in the standard form: ax^2 + bx + c.
Let's solve these equations to find the coefficients:
From Equation 1, we have:
α + β = 6
We can rewrite this equation as: β = 6 - α
Now, substitute β in Equation 2:
α * (6 - α) = 4
Expanding and simplifying the equation:
6α - α^2 = 4
Rearrange the terms:
α^2 - 6α + 4 = 0
This quadratic equation represents the polynomial with α and β as its zeros.
Thus, the polynomial is: P(x) = x^2 - 6x + 4.
Let α + β = 6 , α β = 4
x^2 -( α + β) x + αβ = 0
x^2 - 6 x + 4 = 0 is your polynomial