Form a quadratic polynomial whose sum and product of the zeroes are 3 and 2 respectively. Then find the zeroes and verify the relationship between zeroes and coefficients. Given: α + β = 3, αβ = 2 Polynomial: P(x) = x² - (α + β)x + αβ
Answer (1)
To form a quadratic polynomial with given sum and product of its zeroes, you can use the standard form of a polynomial: P ( x ) = x 2 − ( α + β ) x + α β .
Here, you are given:
α + β = 3
α β = 2
Plug these values into the polynomial formula to construct the quadratic polynomial:
P ( x ) = x 2 − ( 3 ) x + 2
This simplifies to:
P ( x ) = x 2 − 3 x + 2
Next, let's find the zeroes of this polynomial. To find the zeroes, we need to solve the quadratic equation x 2 − 3 x + 2 = 0 .
You can factor this equation:
x 2 − 3 x + 2 = ( x − 1 ) ( x − 2 ) = 0
Setting each factor equal to zero gives the solutions:
x − 1 = 0 \Rightarrow x = 1
x − 2 = 0 \Rightarrow x = 2
Thus, the zeroes of the polynomial are x = 1 and x = 2 .
Verification of the Relationship Between Zeroes and Coefficients:
Sum of Zeroes: α + β = 1 + 2 = 3 This matches the given sum of zeroes.
Product of Zeroes: α β = 1 × 2 = 2 This matches the given product of zeroes.
So, the relationship between the zeroes and the coefficients is verified. The quadratic polynomial formed is x 2 − 3 x + 2 with zeroes 1 and 2, confirming that the given sum and product conditions hold true.
