(3x-1) (7x+2)=0

x= 1/3 or -2/7

P(x)=x^2-{(1/3)+(-2/7)}x-(2/21)=0

Quadratic Polynomial with Zeroes 2alpha and 2beta

To find a quadratic polynomial with zeroes 2alpha and 2beta, we need to understand the relationship between the zeroes of a polynomial and its factors.

Zeroes of a Polynomial
The zeroes of a polynomial are the values of the variable for which the polynomial becomes zero. In other words, if alpha is a zero of a polynomial, then when we substitute alpha in the polynomial, it evaluates to zero.

Factor Theorem
The factor theorem states that if alpha is a zero of a polynomial, then (x - alpha) is a factor of the polynomial. Similarly, if beta is a zero of a polynomial, then (x - beta) is a factor of the polynomial.

Using the factor theorem, we can express the given polynomial as a product of its factors:
P(x) = (x - alpha)(x - beta)

Quadratic Polynomial with Zeroes 2alpha and 2beta
To find a quadratic polynomial with zeroes 2alpha and 2beta, we need to find its factors. Since the zeroes are 2 times alpha and beta, we can write the factors as:
(x - 2alpha)(x - 2beta)

To obtain the quadratic polynomial, we need to expand and simplify the expression:
P(x) = (x - 2alpha)(x - 2beta)
= x^2 - 2beta x - 2alpha x + 4alpha beta
= x^2 - (2beta + 2alpha)x + 4alpha beta

Therefore, the quadratic polynomial with zeroes 2alpha and 2beta is given by:
P(x) = x^2 - (2beta + 2alpha)x + 4alpha beta

Example
Let's consider the given polynomial: 21x^2 - x - 2
If alpha and beta are the zeroes of this polynomial, we can find them by factoring the polynomial or using the quadratic formula.

By factoring, we have:
21x^2 - x - 2 = (3x + 2)(7x - 1)

So, the zeroes are:
alpha = -2/3
beta = 1/7

Now, we can find the quadratic polynomial with zeroes 2alpha and 2beta:
P(x) = x^2 - (2beta + 2alpha)x + 4alpha beta
= x^2 - (2(1/7) + 2(-2/3))x + 4(-2/3)(1/7)
= x^2 - (2/7 - 4/3)x - 8/21

Therefore, the quadratic polynomial with zeroes 2alpha and 2beta is:
P(x) = x^2 - (2/7 - 4/3)x - 8/21